Encoding Method and Device, and Apparatus

ABSTRACT

The present disclosure relates to encoding method and devices. One example method includes determining N to-be-encoded bits, where the N to-be-encoded bits include information bits and frozen bits, obtaining a first polarization weight vector including polarization weights of N polarized channels, where the N to-be-encoded bits correspond to the N polarized channels, determining positions of the information bits based on the first polarization weight vector, and performing polar encoding on the N to-be-encoded bits to obtain polar-encoded bits.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No.PCT/CN2017/090359, filed on Jun. 27, 2017, which claims priority toInternational Application No. PCT/CN2017/088023, filed on Jun. 13, 2017,and Chinese Patent Application No. 201610619696,5, filed on Jul. 29,2016. The disclosures of the aforementioned applications are herebyincorporated by reference in their entireties.

TECHNICAL FIELD

This application relates to the field of encoding and decodingtechnologies, and in particular, to an encoding method and device.

BACKGROUND

Channel encoding is a radio access technology used in a communicationssystem to improve data transmission reliability and ensure communicationquality. A polar (polar) code is an encoding scheme that has lowencoding and decoding complexity and that theoretically proves that aShannon limit can be reached. The polar code is a linear block codewhose generator matrix is G_(N) and whose encoding process is x₁ ^(N)=u₁^(N)G_(N), where x₁ ^(N) is an encoded bit and u₁ ^(N) is ato-be-encoded bit; u₁ ^(N)=(u₁,u₂, . . . ,u_(N)) is a binary row vectorwhose length is N (that is, a code length); and G_(N) is an N×N matrix,and G_(N)=B_(N)F₂ ^(⊗(log) ² ^((N))).

${F_{2} = \begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}},$

and B_(N) is an N×N transposed matrix, for example, a bit reversal (BitReversal) matrix. F₂ ^(⊗(log) ² ^((N))) is defined as a Kronecker(Kronecker) product of log₂ N matrices F₂. In a polar code encodingprocess, positions of information bits and those of frozen bits in u₁^(N) need to be first determined. A bit used to carry information isreferred to as an information bit, and a remaining bit that is set to afixed value pre-agreed on by a transmit end and a receive end isreferred to as a frozen bit. Then, encoding is performed based on x₁^(N)=u₁ ^(N)G^(N). Specifically, polarized channel reliabilitycorresponding to each input bit may be estimated in different manners,and a polarized channel with high reliability is preferentially selectedto place an information bit. In actual application, positions ofinformation bits needs to be indicated by using a sequence, andpositions of information bits may keep the same between a transmit endand a receive end in an online calculation manner or an offline storagemanner.

In the prior art, there are three methods for estimating polarizedchannel reliability. One method is to calculate a Bhattacharyya(Bhattacharyya) parameter of each polarized channel, where the parameterreflects an error probability of the polarized channel, and then toselect polarized channels corresponding to K smallest Bhattacharyyaparameters to place information bits. However, this method is applicableto only a binary erasure channel, and cannot accurately performreliability estimation on other channels, resulting in low performance.The other two methods are a density evolution (Density evolution, DE)method and a Gaussian approximation (Gaussian approximation, GA) method,respectively. The DE method and the GA method have relatively highcalculation complexity and are unsuitable for online calculation. Foroffline storage, the DE method and the GA method depend on a parameterrate matching manner, a code rate, and a modulation scheme of an actualchannel during reliability calculation. A change in any one of theforegoing parameters results in a change in a reliability estimationresult of the polarized channel and a change in a position of acorresponding information bit. This consequently causes excessively highstorage overheads.

SUMMARY

This application provides an encoding method and device, to reducecalculation complexity and storage complexity of polar code encoding.

According to a first aspect, this application provides an encodingmethod, including:

after input information bits are received, determining a quantity N ofto-be-encoded bits based on a target code length M of a polar code,where N is a positive integer; afterwards obtaining a first polarizationweight vector including polarization weights of N polarized channels,where the polarization weights are a basis for information bitselection; then determining positions of the information bits based onthe first polarization weight vector; and finally performing polarencoding on the N to-be-encoded bits to obtain polar-encoded bits.Therefore, positions of the information bits and those of frozen bitsare determined by calculating polarization weights of polar-codepolarized channels, without relating to a channel parameter and a coderate. This can reduce calculation complexity of polar code encoding. Inaddition, polarization weights corresponding to polar codes withdifferent code lengths and code rates have a same part, so that storagespace is greatly reduced, and storage complexity of polar code encodingcan also be reduced.

In a possible design, the target code length M of the polar code isequal to a positive integer power of 2, M is a positive integer, and thedetermining N to-be-encoded bits includes: determining that N is equalto M, where the target code length of the polar code is an output codelength of the polar code; and the obtaining a first polarization weightvector including polarization weights of N polarized channels includes:calculating the polarization weights of the N polarized channels toobtain the first polarization weight vector.

In a possible design, the calculating the polarization weights of the Npolarized channels to obtain the first polarization weight vectorincludes: calculating the polarization weights W_(i) of the N polarizedchannels according to the following formula, to obtain the firstpolarization weight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), n is a positive integer, and ϕ and α areparameters that are preset based on the target code length and a coderate of the polar code.

In a possible design, the calculating the polarization weights of the Npolarized channels to obtain the first polarization weight vectorincludes: obtaining the first polarization weight vector W₀ ^(N−1)through recursion calculation according to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹+β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.

In a possible design, the target code length M of the polar code isequal to a positive integer power of 2, and the obtaining a firstpolarization weight vector including polarization weights of N polarizedchannels includes: obtaining the first polarization weight vector from aprestored second polarization weight vector, where the secondpolarization weight vector includes polarization weights of N_(max)polarized channels, N_(max) is a preset mother code length correspondingto a maximum code length supported by a communications system, and themother code length is a positive integer power of 2.

After the input information bits are received, the quantity N ofto-be-encoded bits is determined based on the target code length M ofthe polar code; afterwards, the first polarization weight vector isobtained from the prestored second polarization weight vector, where thesecond polarization weight vector includes the polarization weights ofthe N_(max) polarized channels, and N_(max) is the mother code lengthcorresponding to the maximum code length supported by the communicationssystem; then, positions of the information bits are determined based onthe first polarization weight vector; and finally, polar encoding isperformed on the N to-be-encoded bits to obtain the polar-encoded bits.Therefore, the position of the information bits and those of the frozenbits are determined by obtaining polarization weights of polar-codepolarized channels in an offline storage manner. For polar codes withdifferent code lengths and code rates, an element set in a polarizationweight vector corresponding to a short-code length is a subset of anelement set in a weight vector corresponding to a long-code length, andpolarized channels with a same index number (sequence number) have asame polarization weight value. Therefore, storage space can be reduced,and storage complexity of polar code encoding can be reduced. Comparedwith online calculation, when there is enough storage space, the offlinestorage manner can shorten latency caused by online calculation.

In a possible design, when the target code length M of the polar code isless than or equal to N_(max), an element set in the first polarizationweight vector is a subset of an element set in the second polarizationweight vector; or when the target code length M of the polar code isgreater than or equal to N_(max), a polarization weight of a polarizedchannel whose channel index number is less than or equal to N_(max) isin a one-to-one correspondence with a polarization weight in the secondpolarization weight vector, and a polarization weight of a polarizedchannel whose channel index number is greater than N_(max) is equal to aweight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), where ϕ andα are parameters that are preset based on the target code length and acode rate of the polar code.

After the input information bits are received, the quantity N ofto-be-encoded bits is determined based on the target code length M ofthe polar code; afterwards, the first polarization weight vector isobtained from the prestored second polarization weight vector, where thesecond polarization weight vector includes the polarization weights ofthe N_(max) polarized channels, and N_(max) is the maximum mother codelength supported by the communications system; then, positions of theinformation bits are determined based on the first polarization weightvector; and finally, polar encoding is performed on the N to-be-encodedbits to obtain the polar-encoded bits. Therefore, the positions of theinformation bits and those of the frozen bits are determined byobtaining polarization weights of polar-code polarized channels in ahalf-online-calculation half-offline-storage manner. For polar codeswith different code lengths and code rates, an element set in apolar-code polarization weight vector corresponding to a code lengthless than prestored N_(max) is a subset of an element set in a weightvector corresponding to N_(max), and polarized channels with a sameindex number (sequence number) have a same polarization weight value;and in a polar-code polarization weight vector corresponding to a codelength greater than prestored N_(max), a polarization weight of apolarized channel whose channel index number is less than or equal toN_(max) is in a one-to-one correspondence with the polarization weightin the second polarization weight vector, and a polarization weight of apolarized channel whose channel index number is greater than N_(max) isequal to a weight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max). Therefore,storage space can be reduced, and storage complexity of polar codeencoding can be reduced. This manner can not only shorten latency causedby online calculation but also reduce storage complexity.

In a possible design, ϕ=0, and α=¼.

In a possible design, the target code length M of the polar code is notequal to a positive integer power of 2, and the determining Nto-be-encoded bits includes: determining that N=2^(└log) ² ^((M−1)┘+1),where the symbol └ ┘ denotes rounding down.

In a possible design, before the obtaining a first polarization weightvector including polarization weights of N polarized channels, themethod further includes: determining positions of N−M to-be-deleted bitsin the N to-be-encoded bits, and calculating, based on a result of thedetermining, a set of modification coefficients including thepolarization weights corresponding to the N to-be-encoded bits, wherethe modification coefficient is used to represent an amount ofmodification to each polarization weight resulting from bit deletion;and the obtaining a first polarization weight vector includingpolarization weights of N polarized channels includes: calculating thepolarization weights of the N polarized channels based on the set ofmodification coefficients, to obtain the first polarization weightvector.

In a possible design, the determining positions of N−M to-be-deletedbits in the N to-be-coded bits includes: determining a channel indexnumber sequence P₀ ^(N−1)=[0, 1, . . . , N−1] of the N polarizedchannels; and performing the following operations on each index numberin the channel index number sequence; performing bit reversal on abinary number corresponding to each index number, and converting, to adecimal number, a binary number obtained after bit reversal, to obtain anew channel index number sequence P₀ ^(N−1)′; sorting index numbers inthe new channel index number sequence P₀ ^(N−1)′; and selecting, inascending order, N−M index numbers in the channel index number sequenceP₀ ^(N−1)′ as index numbers of the to-be-deleted bits.

It should be noted that in this step, an operation performed on eachindex number is not limited to a bit reversal operation, for example,interleaving, scrambling, or the like may be performed on the binarynumber corresponding to each index number, or no processing is performedon each index number.

In a possible design, the set of modification coefficients is

${g = \begin{bmatrix}g_{({0,0})} & \; & \ldots & \; & g_{({0,{n - 1}})} \\g_{({1,0})} & \; & \; & \; & g_{({1,{n - 1}})} \\\vdots & \; & \vdots & \; & \vdots \\g_{({{N - 1},0})} & \; & \; & \; & g_{({{N - 1},{n - 1}})} \\g_{({N,0})} & g_{({N,1})} & \; & \ldots & g_{({N,{n - 1}})}\end{bmatrix}};$

and

the calculating, based on a result of the determining, a set ofmodification coefficients including modification coefficients of the Nbits includes: calculating each item g_((i,j)) in the set g ofmodification coefficients according to the following formulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)), where

iϵ{0,1, . . . ,N−1}, jϵ{0,1, . . . ,n−1}, N=2^(n), P_(i) is anidentifier indicating whether a bit corresponding to a polarized channelwhose channel index number is i is to be deleted, and P_(i)ϵ{0,1}.

In a possible design, the calculating the polarization weights of the Npolarized channels based on the set of modification coefficients, toobtain the first polarization weight vector includes: calculating thepolarization weights W_(i) of the N polarized channels according to thefollowing formula, to obtain the first polarization weight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {1 - g_{({i,j})}} )*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), g_((i,j)) is an element in row i and column j inthe set g of modification coefficients, and ϕ and α are parameters thatare preset based on the target code length and a code rate of the polarcode.

In a possible design, the calculating the polarization weights of the Npolarized channels based on the set of modification coefficients, toobtain the first polarization weight vector includes: obtaining thefirst polarization weight vector W₀ ^(N−1) through recursion calculationaccording to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹ +g _((i,j))*β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, g_((i,j)) isan element in row i and column j in the set g of modificationcoefficients, and W₀ is an arbitrary constant.

In the foregoing possible design, after the input information bits arereceived, the quantity N of to-be-encoded bits is determined based onthe target code length M of the polar code; afterwards, the positions ofthe N−M to-be-deleted bits in the N to-be-encoded bits are determined,and the set of modification coefficients including the modificationcoefficients of the N to-be-encoded bits is calculated based on theresult of the determining; then, the polarization weights of the Npolarized channels are calculated based on the set of modificationcoefficients, to obtain the first polarization weight vector; andfinally, positions of the information bits are determined based on thefirst polarization weight vector, and polar encoding is performed on theN to-be-encoded bits to obtain the polar-encoded bits. Therefore, for apolar code with any target code length, positions of information bitsand those of frozen bits are determined by calculating polarizationweights of polar-code polarized channels, without relating to a channelparameter and a code rate. This can reduce calculation andimplementation complexity of polar code encoding.

In a possible design, ϕ=0, and α=¼.

In a possible design, the determining positions of the information bitsbased on the first polarization weight vector includes: sorting thepolarization weights in the first polarization weight vector indescending order, and selecting bits corresponding to the first Kpolarization weights as information bits.

In a possible design, the encoding method is executed by a base stationor a terminal.

According to a second aspect, this application provides an encodingdevice, including:

a first determining module, configured to determine N to-be-encodedbits, where the N to-be-encoded bits include information bits and frozenbits, and N is a positive integer; an obtaining module, configured toobtain a first polarization weight vector including polarization weightsof N polarized channels, where the polarization weights are a basis forinformation bit selection, and the N to-be-encoded bits arecorresponding to the N polarized channels; a second determining module,configured to determine positions of the information bits based on thefirst polarization weight vector; and an encoding module, configured toperform polar encoding on the N to-be-encoded bits to obtainpolar-encoded bits.

In a possible design, a target code length M of a polar code is equal toa positive integer power of 2, M is a positive integer, and the firstdetermining module is specifically configured to determine that N isequal to M, where the target code length of the polar code is an outputcode length of the polar code; and the obtaining module is specificallyconfigured to calculate the polarization weights of the N polarizedchannels to obtain the first polarization weight vector.

In a possible design, the obtaining module is specifically configured tocalculate the polarization weights W_(i) of the N polarized channelsaccording to the following formula, to obtain the first polarizationweight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), n is a positive integer, and ϕ and α areparameters that are preset based on the target code length and a coderate of the polar code.

In a possible design, the obtaining module is specifically configured toobtain the first polarization weight vector W₀ ^(N−1) through recursioncalculation according to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹ +g _((i,j))*β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.

In a possible design, a target code length M of a polar code is equal toa positive integer power of 2, and the obtaining module is specificallyconfigured to obtain the first polarization weight vector from aprestored second polarization weight vector, where the secondpolarization weight vector includes polarization weights of N_(max)polarized channels, N_(max) is a preset mother code length correspondingto a maximum code length supported by a communications system, and themother code length is a positive integer power of 2.

In a possible design, when the target code length M of the polar code isless than or equal to N_(max), an element set in the first polarizationweight vector is a subset of an element set in the second polarizationweight vector; or when the target code length M of the polar code isgreater than or equal to N_(max), a polarization weight of a polarizedchannel whose channel index number is less than or equal to N_(max), isin a one-to-one correspondence with a polarization weight in the secondpolarization weight vector, and a polarization weight of a polarizedchannel whose channel index number is greater than N_(max) is equal to aweight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), where ϕ andα are parameters that are preset based on the target code length and acode rate of the polar code.

In a possible design, ϕ=0, and α=¼.

In a possible design, a target code length M of a polar code is notequal to a positive integer power of 2, and the first determining moduleis specifically configured to determine that N=2^(└log) ² ^((M−1)┘+1),where the symbol └ ┘ denotes rounding down.

In a possible design, the encoding device further includes: a thirddetermining module, configured to: before the obtaining module obtainsthe first polarization weight vector including the polarization weightsof the N polarized channels, determine positions of N−M to-be-deletedbits in the N to-be-encoded bits; and a calculation module, configuredto calculate, based on a result of the determining performed by thethird determining module, a set of modification coefficients includingthe polarization weights corresponding to the N to-be-encoded bits,where the modification coefficient is used to represent an amount ofmodification to each polarization weight resulting from bit deletion,where the obtaining module is specifically configured to calculate thepolarization weights of the N polarized channels based on the set ofmodification coefficients, to obtain the first polarization weightvector.

In a possible design, the third determining module is specificallyconfigured to: determine a channel index number sequence P₀ ^(N−1)=[0,1, . . . , N−1] of the N polarized channels; and perform the followingoperations on each index number in the channel index number sequence:performing bit reversal on a binary number corresponding to each indexnumber, and converting, to a decimal number, a binary number obtainedafter bit reversal, to obtain a new channel index number sequence P₀^(N−1)′; sorting index numbers in the new channel index number sequenceP₀ ^(N−1)′; and selecting, in ascending order, N−M index numbers in thechannel index number sequence P₀ ^(N−1)′ as index numbers of theto-be-deleted bits.

In a possible design, the set of modification coefficients is

${g = \begin{bmatrix}g_{({0,0})} & \; & \ldots & \; & g_{({0,{n - 1}})} \\g_{({1,0})} & \; & \; & \; & g_{({1,{n - 1}})} \\\vdots & \; & \vdots & \; & \vdots \\g_{({{N - 1},0})} & \; & \; & \; & g_{({{N - 1},{n - 1}})} \\g_{({N,0})} & g_{({N,1})} & \; & \ldots & g_{({N,{n - 1}})}\end{bmatrix}};$

and the calculation module is specifically configured to calculate eachitem g_((i,j)) in the set g of modification coefficients according tothe following formulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)), where

iϵ{0,1, . . . ,N−1}, jϵ{0,1, . . . ,n−1}, N=2^(n), P_(i) is anidentifier indicating whether a bit corresponding to a polarized channelwhose channel index number is i is to be deleted, and P_(i)ϵ{0,1}.

In a possible design, the obtaining module is specifically configured tocalculate the polarization weights W_(i) of the N polarized channelsaccording to the following formula, to obtain the first polarizationweight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {1 - g_{({i,j})}} )*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), g_((i,j)) is an element in row i and column j inthe set g of modification coefficients, and ϕ and α are parameters thatare preset based on the target code length and a code rate of the polarcode.

In a possible design, the obtaining module is specifically configured toobtain the first polarization weight vector W₀ ^(N−1) through recursioncalculation according to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹ +g _((i,j))*β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, g_((i,j)) isan element in row i and column j in the set g of modificationcoefficients, and W₀ is an arbitrary constant.

In a possible design, ϕ=0, and α=14.

In a possible design, the second determining module is specificallyconfigured to: sort the polarization weights in the first polarizationweight vector in descending order, and select bits corresponding to thefirst K polarization weights as information bits.

In a possible design, the encoding device is a base station or aterminal.

For beneficial effects of the encoding device provided in the secondaspect and all the possible designs of the second aspect, refer tobeneficial effects brought by the first aspect and all the possibledesigns of the first aspect, and details are not repeated herein.

According to a third aspect, this application provides a communicationsapparatus, including:

a memory, configured to store an instruction for performing anoperation; and

a processor, coupled with the memory, and according to the instruction,configured to: determine N to-be-encoded bits, where the N to-be-encodedbits include information bits and frozen bits, and N is a positiveinteger; obtain a first polarization weight vector includingpolarization weights of N polarized channels, where the polarizationweights are a basis for information bit selection, and the Nto-be-encoded bits are corresponding to the N polarized channels;determine positions of the information bits based on the firstpolarization weight vector; and perform polar encoding on the Nto-be-encoded bits to obtain polar-encoded bits.

In a possible design, the communications apparatus is a base station ora terminal.

According to a fourth aspect, this application provides a computerstorage medium, where the computer storage medium stores acomputer-executable instruction, and when the computer-executableinstruction is invoked by a computer, the computer executes the methodaccording to all the foregoing embodiments.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic diagram of an existing polar code encodingprocess:

FIG. 2 is a block diagram of an encoding structure in Embodiment 1 of anencoding method according to this application:

FIG. 3 is a flowchart of Embodiment 1 of an encoding method according tothis application:

FIG. 4 is a flowchart of Embodiment 2 of an encoding method according tothis application:

FIG. 5 is a schematic diagram of polarization weight vectorsrespectively corresponding to polar codes with code lengths N being 4,8, and 16 according to Embodiment 2 of this application;

FIG. 6 is a schematic diagram of polarization weight vectorsrespectively corresponding to polar codes with code lengths N being 4,8, and 16 according to Embodiment 3 of this application;

FIG. 7 is a flowchart of Embodiment 4 of an encoding method according tothis application:

FIG. 8 is a schematic structural diagram of Embodiment 1 of an encodingdevice according to this application;

FIG. 9 is a schematic structural diagram of Embodiment 2 of an encodingdevice according to this application; and

FIG. 10 is a schematic structural diagram of an embodiment of acommunications apparatus according to this application.

DESCRIPTION OF EMBODIMENTS

FIG. 1 is a schematic diagram of an existing polar code encodingprocess. Polar code encoding is performed by a transmit end. As shown inFIG. 1, a polar code encoding process includes three processingprocesses: construction, encoding, and rate matching. First, in theconstruction process, after input data (namely, input information bits,for example, N₁ information bits) is received, a quantity N ofto-be-encoded bits needs to be determined based on a target code lengthM obtained after encoding, where the to-be-encoded bits includeinformation bits (whose quantity is N₁) and frozen bits (whose quantityis N₂), and N=N₁+N₂; and positions of the information bits and those ofthe frozen bits need to be determined, where one bit is corresponding toone polarized channel. Then, in the encoding process, polar encoding isperformed on the to-be-encoded bits to obtain polar-encoded bits.Finally, in the rate matching process, because a code length obtainedafter polar code encoding is a positive integer power of 2, and duringactual implementation, the code length needs to be flexibly configured,rate matching needs to be performed to obtain a code with any codelength. A specific implementation is deleting some N−M bits from a codewith the code length of a positive integer power of 2, to obtain a codewith a target code length. N, M, N₁, and N₂ are all positive integers.An encoding method provided in this application mainly relates to thepolar code encoding process shown in FIG. 1, and mainly relates to theconstruction process. In an existing construction process, in the priorart, channel reliability estimation is performed mostly based on actualparameters and code rates of all polarized channels, so as to determinepositions of information bits, and remaining positions are positions offrozen bits. In actual application, the positions of the informationbits need to be indicated by using a sequence, and positions ofinformation bits may keep the same in an online calculation manner or anoffline storage manner. During offline storage, a quite large table isrequired to store the sequence, and due to different channel parameters,code rates, and code lengths, millions of sequences need to be stored.Therefore, in an existing manner of determining positions of informationbits, both online calculation and offline storage are relativelycomplex. In view of the foregoing problem, this application provides anencoding method. Positions of information bits and those of frozen bitsare determined by calculating polarization weights of polar-codepolarized channels, without relating to a channel parameter and a coderate. This can reduce calculation complexity and storage complexity ofpolar code encoding. The following describes in detail the encodingmethod and device provided in this application with reference to theaccompanying drawings.

This application is mainly applied to various wireless communicationssystems, and acts on digital signal processing units in the systems, toimprove reliability of wireless communication.

Network elements in this application are mainly a base station and aterminal. The base station implements communication with the terminal.

FIG. 2 is a block diagram of an encoding structure in Embodiment 1 of anencoding method according to this application. As shown in FIG. 2, bothinformation bits (whose quantity is N₁) and a target code length M of apolar code are inputs, and then polar code construction is performed. Nto-be-encoded bits are first determined based on the target code lengthM of the polar code. A quantity of frozen bits is N₂, and N=N₁+N₂.Positions of the information bits and those of the frozen bits are thendetermined. Then, encoding is performed by using an encoding matrixG_(N), to obtain polar-encoded bits.

FIG. 3 is a flowchart of Embodiment 1 of an encoding method according tothis application. As shown in FIG. 3, the method in this embodiment mayinclude the following steps.

S101: Determine N to-be-encoded bits, where the N to-be-encoded bitsinclude information bits and frozen bits.

Specifically, a code length obtained after polar code encoding is apositive integer power of 2, and during actual implementation, the codelength needs to be flexibly configured, in other words, a polar codewith any code length needs to be obtained. Therefore, after inputinformation bits are received, the quantity N of to-be-encoded bitsneeds to be determined based on a target code length M of a polar code.The N to-be-encoded bits include the information bits and the frozenbits. The target code length of the polar code is an output code lengthof the polar code. Specifically, there are two cases. One case is thatthe target code length M of the polar code is equal to a positiveinteger power of 2, to be specific, there is no need to perform ratematching. In this case, the determined quantity N of to-be-encoded bitsis equal to M. The other case is that the target code length M of thepolar code is not equal to a positive integer power of 2, to bespecific, rate matching needs to be performed. In this case, thedetermined quantity of to-be-encoded bits is N=2^(└log) ² ^((M−1)┘+1),where the symbol └ ┘ denotes rounding down.

S102: Obtain a first polarization weight vector including polarizationweights of N polarized channels, where the polarization weights are abasis for information bit selection, and the N to-be-encoded bits arecorresponding to the N polarized channels.

Specifically, one bit is corresponding to one polarized channel, and apolarization weight of the polarized channel is a basis for informationbit selection. The polarization weight is different from channelreliability in the prior art. For a case in which the target code lengthM of the polar code is equal to a positive integer power of 2 and a casein which the target code length M of the polar code is not equal to apositive integer power of 2, manners of obtaining the polarizationweights are different. The following describes in detail how to obtainthe first polarization weight vector when the quantity N ofto-be-encoded bits is equal to the target code length M of the polarcode. Specifically, the polarization weights W_(i) of the N polarizedchannels in the first polarization weight vector are calculatedaccording to the following formula:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), and ϕ and α are parameters that are preset basedon the target code length and a code rate of the polar code. For polarcodes with different code lengths and code rates, the parameters ϕ and αmay be adjusted, to obtain an information bit indication sequence of thepolar code for better performance.

Optionally, in

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$

ϕ=0, α=¼, and therefore the calculation formula is

$W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*{2^{\frac{j}{4}}.}}}$

The polarization weight of the polarized channel is calculated accordingto the formula, to determine the information bit, to be specific, toobtain the information bit indication sequence for better performance.For example, if N=8, n=log₂(8)=3. For i=3 (i

011), a process of calculating a polarization weight W₃ is as follows:

W ₃=1×2̂(0×(¼))+1×2̂(1×(¼))+0×2̂(2×(¼))=2.8192.

It should be noted that corresponding polarization weights in all theembodiments of this application are corresponding to a polar-codeencoding matrix F_(N) excluding a bit reversal operation B_(N). If theencoding matrix includes the bit reversal operation B_(N), acorresponding bit reversal operation needs to be performed on thecalculated polarization weights.

In another implementation, the first polarization weight vector W₀^(N−1) is obtained through recursion calculation according to thefollowing formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹+β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.

It can be learned from the foregoing formulas that a relationshipbetween polarization weights of polar codes in two adjacent orders maybe simply described as W^(N)=[W^(N/2), W^(N/2)+β_(i)], where i=log₂(N).When i=1, the value is substituted into the foregoing recursioncalculation formulas, to obtain initial formulas for recursioncalculation: β₁=(1+ϕ)^(α). W₁=W₀+β₁, and

W ₀ ¹=[W ₀ ,W ₁].

For example, when W₀=0, a calculation formula

$W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}$

may be obtained. When W₀=0, ϕ=0, and α=¼, a calculation formula

$W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*2^{\frac{j}{4}}}}$

may be obtained.

The following uses N=8 as an example, and a recursion calculationprocess is as follows:

First, a polarization weight of a polarized channel is initialized asW₀=a, where a is a constant value, and usually may be set to 0. Then,based on a code length N (N=2^(n)) of the polar code, i is set to 1 ton, and the polarization weight of the polarized channel increases orderby order to the target code length N, and the calculation process isshown as follows:

-   -   (1) Set an initial value: W₀=a.    -   (2) i=1, β₁=(1+ϕ)^(α), W₁=W₀+β₁, and W₀ ¹=[W₀,W₁]=[W₀,W₀+β₁].    -   (3) i=2, β₂=(2+ϕ)^(α), and [W₀, W₁, W₂, W₃]=[W₀, W₁, W₀+β₂,        W₁+β₂].    -   (4) i=3, β₃=(4+ϕ)^(α), and [W₀, W₁, W₂, W₃, W₄, W₅, W₆, W₇]=[W₀,        W₁, W₂, W₃, W₀+β₃, W₁+β₃, W₂+β₃, W₃+β₃]    -   Finally, W₀ ⁷=[W₀, W₁, W₂, W₃, W₄, W₅, W₆, W₇] is obtained.

S103: Determine positions of the information bits based on the firstpolarization weight vector.

Specifically, the polarization weights in the first polarization weightvector may be sorted in descending order, and bits corresponding to thefirst K polarization weights are selected as information bits. To bespecific, positions of the information bits are selected, and remainingpositions are set for the frozen bits, so as to obtain the completeto-be-encoded bits with a length of N.

S104: Perform polar encoding on the N to-be-encoded bits to obtainpolar-encoded bits.

Specifically, the polar-code encoding matrix F_(N) is used to completean encoding process of the to-be-encoded bits.

According to the encoding method provided in this embodiment, after theinput information bits are received, the quantity N of to-be-encodedbits is determined based on the target code length M of the polar code;afterwards, the first polarization weight vector including thepolarization weights of the N polarized channels is obtained; then,positions of the information bits are determined based on the firstpolarization weight vector; and finally, polar encoding is performed onthe N to-be-encoded bits to obtain the polar-encoded bits. Therefore,the positions of the information bits and those of the frozen bits aredetermined by calculating polarization weights of polar-code polarizedchannels, without relating to a channel parameter and a code rate. Thiscan reduce calculation complexity of polar code encoding. In addition,polarization weights corresponding to polar codes with different codelengths and code rates have a same part, so that storage space isgreatly reduced, and storage complexity of polar code encoding can alsobe reduced.

Still further, it can be learned, according to the calculation formulaof the polarization weights in Embodiment 1 shown in FIG. 3, that adifference between polarization weights of polar codes in two adjacentorders is a weight increment β_(i). In a specific implementation of S102in Embodiment 1, the first polarization weight vector is obtainedthrough online calculation. In another implementation, the firstpolarization weight vector may be obtained in an offline storage manner.Compared with online calculation, when there is enough storage space,the offline storage manner can shorten latency caused by onlinecalculation. The following describes in detail the offline storagemanner with reference to FIG. 4. It should be noted that the offlinestorage manner is applicable to a scenario in which a target code lengthM of a polar code is equal to a positive integer power of 2.

FIG. 4 is a flowchart of Embodiment 2 of an encoding method according tothis application. As shown in FIG. 4, the method in this embodiment mayinclude the following steps.

S201: Determine N to-be-encoded bits, where the N to-be-encoded bitsinclude information bits and frozen bits.

S202: Obtain a first polarization weight vector from a prestored secondpolarization weight vector, where an element set in the firstpolarization weight vector is a subset of an element set in the secondpolarization weight vector, the second polarization weight vectorincludes polarization weights of N_(max) polarized channels, N_(max) isa preset mother code length corresponding to a maximum code lengthsupported by a communications system, and the mother code length is apositive integer power of 2.

Specifically, the mother code length N_(max) corresponding to themaximum code length supported by the communications system may be presetbased on an actual communication scenario, where the mother code lengthis a positive integer power of 2; and the second polarization weightvector is calculated in advance by using the calculation formulas shownin Embodiment 1, and is stored offline. During usage, the firstpolarization weight vector is directly obtained from the prestoredsecond polarization weight vector. It can be learned from thecalculation formulas shown in Embodiment 1 that when a code length is apositive integer power of 2, an element set in a polarization weightvector corresponding to a short-code length is a subset of an elementset in a weight vector corresponding to a long-code length, andpolarized channels with a same index number (sequence number) have asame polarization weight value.

An example of polar codes with code lengths N being 4, 8, and 16 isused. FIG. 5 is a schematic diagram of polarization weight vectorsrespectively corresponding to polar codes with code lengths N being 4,8, and 16 according to Embodiment 2 of this application. As shown inFIG. 5, an element set in the polarization weight vector correspondingto the code length N=8 is a subset of an element set in the polarizationweight vector corresponding to the code length N=16, and an element setin the polarization weight vector corresponding to the code length N=4is a subset of an element set in the polarization weight vectorcorresponding to the code length N=8. In addition, polarization weightsof polarized channels are the same in a same position. Therefore, apolarization weight vector, corresponding to a maximum code lengthN_(max), of polarized channels is prestored, so that a polarized channelweight vector corresponding to any code length N less than or equal toN_(max) can be obtained.

S203: Determine positions of the information bits based on the firstpolarization weight vector.

Specifically, the polarization weights in the first polarization weightvector may be sorted in descending order, and bits corresponding to thefirst K polarization weights are selected as information bits. To bespecific, positions of the information bits are selected, and remainingpositions are set for the frozen bits, so as to obtain the completeto-be-encoded bits with a length of N.

S204: Perform polar encoding on the N to-be-encoded bits to obtainpolar-encoded bits.

Specifically, a polar-code encoding matrix F_(N) is used to complete anencoding process of the to-be-encoded bits.

According to the encoding method provided in this embodiment, afterinput information bits are received, the quantity N of to-be-encodedbits is determined based on the target code length M of the polar code;afterwards, the first polarization weight vector is obtained from theprestored second polarization weight vector, where the secondpolarization weight vector includes the polarization weights of theN_(max) polarized channels, and N_(max) is the mother code lengthcorresponding to the maximum code length supported by the communicationssystem; then, positions of the information bits are determined based onthe first polarization weight vector; and finally, polar encoding isperformed on the N to-be-encoded bits to obtain the polar-encoded bits.Therefore, the positions of the information bits and those of the frozenbits are determined by obtaining polarization weights of polar-codepolarized channels in an offline storage manner. For polar codes withdifferent code lengths and code rates, an element set in a polarizationweight vector corresponding to a short-code length is a subset of anelement set in a weight vector corresponding to a long-code length, andpolarized channels with a same index number (sequence number) have asame polarization weight value. Therefore, storage space can be reduced,and storage complexity of polar code encoding can be reduced. Comparedwith online calculation, when there is enough storage space, the offlinestorage manner can shorten latency caused by online calculation.

Optionally, in a specific implementation different from that in S202 inEmbodiment 2, the first polarization weight vector is obtained in theoffline storage manner. In another implementation, the firstpolarization weight vector may be obtained in a half-online-calculationhalf-offline-storage manner. The following describes in detail thehalf-online-calculation half-offline-storage manner in Embodiment 3 ofthis application. It should be noted that this manner is applicable to ascenario in which a target code length M of a polar code is equal to apositive integer power of 2.

The obtaining a first polarization weight vector from a prestored secondpolarization weight vector in S202 may be specifically: When the targetcode length M of the polar code is less than or equal to N_(max), theelement set in the first polarization weight vector is the subset of theelement set in the second polarization weight vector; or when the targetcode length M of the polar code is greater than or equal to N_(max), apolarization weight of a polarized channel whose channel index number isless than or equal to N_(max) is in a one-to-one correspondence with apolarization weight in the second polarization weight vector, and apolarization weight of a polarized channel whose channel index number isgreater than N_(max) is equal to a weight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), where ϕ andα are parameters that are preset based on the target code length and acode rate of the polar code. For example, FIG. 6 is a schematic diagramof polarization weight vectors respectively corresponding to polar codeswith code lengths N being 4, 8, and 16 according to Embodiment 3 of thisapplication. In this embodiment, N_(max)=8. As shown in FIG. 6,polarization weights stored in the second polarization weight vector areW0 to W7. Polarization weights corresponding to code lengths less thanN_(max)=8 are extracted from the second polarization weight vector, andpolarization weights corresponding to index numbers 8 to 15 in apolarization weight vector corresponding to a code length 16 areobtained by adding a weight increment (8+ϕ)^(α) to the polarizationweights corresponding to index numbers 0 to 7.

Optionally, in the foregoing embodiment, ϕ=0, and α=¼.

According to the encoding method provided in this embodiment, afterinput information bits are received, the quantity N of to-be-encodedbits is determined based on the target code length M of the polar code;afterwards, the first polarization weight vector is obtained from theprestored second polarization weight vector, where the secondpolarization weight vector includes the polarization weights of theN_(max) polarized channels, and N_(max) is the maximum mother codelength supported by the communications system: then, positions of theinformation bits are determined based on the first polarization weightvector; and finally, polar encoding is performed on the N to-be-encodedbits to obtain the polar-encoded bits. Therefore, the positions of theinformation bits and those of the frozen bits are determined byobtaining polarization weights of polar-code polarized channels in thehalf-online-calculation half-offline-storage manner. For polar codeswith different code lengths and code rates, an element set in apolar-code polarization weight vector corresponding to a code lengthless than prestored N_(max) is a subset of an element set in a weightvector corresponding to N_(max), and polarized channels with a sameindex number (sequence number) have a same polarization weight value;and in a polar-code polarization weight vector corresponding to a codelength greater than prestored N_(max), a polarization weight of apolarized channel whose channel index number is less than or equal toN_(max) is in a one-to-one correspondence with the polarization weightin the second polarization weight vector, and a polarization weight of apolarized channel whose channel index number is greater than N_(max) isequal to a weight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max). Therefore,storage space can be reduced, and storage complexity of polar codeencoding can be reduced. This manner can not only shorten latency causedby online calculation but also reduce storage complexity.

FIG. 7 is a flowchart of Embodiment 4 of an encoding method according tothis application. In this embodiment, the following describes in detailhow to obtain a first polarization weight vector when a target codelength M of a polar code is not equal to a positive integer power of 2,to be specific, when rate matching needs to be performed. As shown inFIG. 7, the method in this embodiment may include the following steps.

S301: Determine N to-be-encoded bits, where the N to-be-encoded bitsinclude information bits and frozen bits.

Specifically, a code length obtained after polar code encoding is apositive integer power of 2, and during actual implementation, the codelength needs to be flexibly configured, in other words, a polar codewith any code length needs to be obtained. Therefore, after inputinformation bits are received, the quantity N of to-be-encoded bitsneeds to be determined based on a target code length M of a polar code.The N to-be-encoded bits include the information bits and the frozenbits. When the target code length M of the polar code is not equal to apositive integer power of 2, to be specific, rate matching needs to beperformed, the determined quantity of to-be-encoded bits is N=2^(└log) ²^((M−1)┘+1), where the symbol └ ┘ denotes rounding down.

S302: Determine positions of N−M to-be-deleted bits in the Nto-be-encoded bits, and calculate, based on a result of the determining,a set of modification coefficients including polarization weightscorresponding to the N to-be-encoded bits, where the modificationcoefficient is used to represent an amount of modification to eachpolarization weight resulting from bit deletion.

Specifically, the determining positions of N−M to-be-deleted bits in theN to-be-encoded bits includes the following steps:

(1) Determine a channel index number sequence P₀ ^(N−1)=[0,1, . . .,N−1] of N polarized channels.

(2) Perform the following operations on each index number in the channelindex number sequence:

(3) Perform bit reversal on a binary number corresponding to each indexnumber, and convert, to a decimal number, a binary number obtained afterbit reversal, to obtain a new channel index number sequence P₀ ^(N−1)′.

It should be noted that in this step, an operation performed on eachindex number is not limited to a bit reversal operation, for example,interleaving, scrambling, or the like may be performed on the binarynumber corresponding to each index number, or no processing is performedon each index number.

(4) Sort index numbers in the new channel index number sequence P₀^(N−1)′.

(5) Select, in ascending order, N−M index numbers in the channel indexnumber sequence P₀ ^(N−1)′ as index numbers of the to-be-deleted bits.

For example, in the original channel index number sequence P₀ ^(N−1)=[0,1, . . . , N−1], a binary number corresponding to an index number in theoriginal channel index number sequence is A_(i)

B_(n−1)B_(n−2) . . . B₀, and a binary number obtained after bit reversalis A′_(i)

B₀B₁ . . . B_(n−1), where B_(j)ϵ{0,1} and jϵ{0, 1, . . . , n−1}. Afterthe bit reversal operation is performed on all elements in P₀ ^(N−1), P₀^(N−1)′ is obtained. After all the index numbers in P₀ ^(N−1)′ arere-sorted, bits corresponding to N−M index numbers are selected from thechannel index number sequence P₀ ^(N−1)′ in ascending order asto-be-deleted bits.

Due to introduction of bit deletion, coefficients of the polarizationweights need to be modified, and the corresponding set g of modificationcoefficients is in a one-to-one correspondence with each computing nodein a butterfly decoding tree, and therefore is an N×n vector. The set ofmodification coefficients is

${g = \begin{bmatrix}g_{({0,0})} & {\; \ldots} & \; & g_{({0,{n - 1}})} \\g_{({1,0})} & \; & \; & g_{({1,{n - 1}})} \\\vdots & {\; \vdots} & \; & \vdots \\g_{({{N - 1},0})} & \; & \; & g_{({{N - 1},{n - 1}})} \\g_{({N,0})} & g_{({N,1})} & \ldots & g_{({N,{n - 1}})}\end{bmatrix}};$

and

the calculating, based on a result of the determining, a set ofmodification coefficients including modification coefficients of the Nbits may be specifically: calculating each item g_((i,j)) in the set gof modification coefficients according to the following formulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)), where

iϵ{0,1, . . . ,N−1}, jϵ{0,1, . . . ,n−1}, N=2^(n), P_(i) is anidentifier indicating whether a bit corresponding to a polarized channelwhose channel index number is i is to be deleted, and P_(i)ϵ{0,1}.

Specifically, when j=n−1, initial g is calculated by using the followingformulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)); and

when I=[n−2, n−3, . . . , 1,0], remaining coefficients of g arecalculated by using the following formulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)).

The following provides an example for description. For example, N=2^(n):

-   -   when j=2,

g _((0,2))=(P ₀ +P ₄)/2, and g _((4,2)) =g _((0,2));

g _((1,2))=(P ₁ +P ₅)/2, and g _((5,2)) =g _((1,2));

g _((2,2))=(P ₂ +P ₆)/2, and g _((6,2)) =g _((2,2)); and

g _((3,2))=(P ₃ +P ₇)/2, and g _((7,2)) =g _((3,2));

when jϵ{0,1}

g _((0,1))=(g _((0,2)) +g _((2,2)))/2, and g _((2,1)) =g _((0,1));

g _((1,1))=(g _((1,2)) +g _((3,2)))/2, and g _((3,1)) =g _((1,1)); and

g _((0,0))=(g _((0,1)) +g _((1,1)))/2, and g _((1,0)) =g _((0,1));

Remaining g_((i,j)) is calculated according to the same rule.

S303: Calculate polarization weights of N polarized channels based onthe set of modification coefficients, to obtain a first polarizationweight vector.

Specifically, in an implementation, the polarization weights W₁ of the Npolarized channels are calculated according to the following formula, toobtain the first polarization weight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {1 - g_{({i,j})}} )*( {2^{j} + \varphi} )^{\alpha}}}},$

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), g_((i,j)) is an element in row i and column j inthe set g of modification coefficients, and ϕ and α are parameters thatare preset based on the target code length and a code rate of the polarcode.

In another implementation, the first polarization weight vector W₀^(N−1) is obtained through recursion calculation according to thefollowing formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹ +g _((i,j))*β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, i, ϕ and α are parameters that are preset basedon the target code length and a code rate of the polar code, g_((i,j))is an element in row i and column j in the set g of modificationcoefficients, and W₀ is an arbitrary constant.

Optionally, in the foregoing formula, 4=0, and α=¼.

S304: Determine positions of the information bits based on the firstpolarization weight vector.

Specifically, the polarization weights in the first polarization weightvector may be sorted in descending order, and bits corresponding to thefirst K polarization weights are selected as information bits. To bespecific, positions of the information bits are selected, and remainingpositions are set for the frozen bits, so as to obtain the completeto-be-encoded bits with a length of N.

S305: Perform polar encoding on the N to-be-encoded bits to obtainpolar-encoded bits.

Specifically, a polar-code encoding matrix F_(N) is used to complete anencoding process of the to-be-encoded bits.

According to the encoding method provided in this embodiment, after theinput information bits are received, the quantity N of to-be-encodedbits is determined based on the target code length M of the polar code;afterwards, the positions of the N−M to-be-deleted bits in the Nto-be-encoded bits are determined, and the set of modificationcoefficients including the modification coefficients of the Nto-be-encoded bits is calculated based on the result of the determining;then, the polarization weights of the N polarized channels arecalculated based on the set of modification coefficients, to obtain thefirst polarization weight vector; and finally, positions of theinformation bits are determined based on the first polarization weightvector, and polar encoding is performed on the N to-be-encoded bits toobtain the polar-encoded bits. Therefore, for a polar code with anytarget code length, positions of information bits and those of frozenbits are determined by calculating polarization weights of polar-codepolarized channels, without relating to a channel parameter and a coderate. This can reduce calculation and implementation complexity of polarcode encoding.

FIG. 8 is a schematic structural diagram of Embodiment 1 of an encodingdevice according to this application. As shown in FIG. 8, the encodingdevice 100 in this embodiment may include a first determining module 11,an obtaining module 12, a second determining module 13, and an encodingmodule 14. The first determining module 11 is configured to determine Nto-be-encoded bits. The N to-be-encoded bits include information bitsand frozen bits. The obtaining module 12 is configured to obtain a firstpolarization weight vector including polarization weights of N polarizedchannels. The polarization weights are a basis for information bitselection, and the N to-be-encoded bits are corresponding to the Npolarized channels. The second determining module 13 is configured todetermine positions of the information bits based on the firstpolarization weight vector. The encoding module 14 is configured toperform polar encoding on the N to-be-encoded bits to obtainpolar-encoded bits.

Further, a target code length M of a polar code is equal to a positiveinteger power of 2, and the first determining module 11 is specificallyconfigured to determine that N is equal to M. The target code length ofthe polar code is an output code length of the polar code. The obtainingmodule 12 is specifically configured to calculate the polarizationweights of the N polarized channels to obtain the first polarizationweight vector.

Optionally, the obtaining module 12 is specifically configured tocalculate the polarization weights W_(i) of the N polarized channelsaccording to the following formula, to obtain the first polarizationweight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code.

Optionally, the obtaining module 12 is specifically configured to obtainthe first polarization weight vector W₀ ^(N−1) through recursioncalculation according to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹+β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.

The encoding device in this embodiment may be configured to execute thetechnical solution in the method embodiment shown in FIG. 3. Theirimplementation principles are similar, and details are not repeatedherein.

It should be noted that the encoding device in this application may be abase station or a terminal.

After receiving input information bits, the encoding device provided inthis embodiment determines the quantity N of to-be-encoded bits based onthe target code length M of the polar code; afterwards obtains the firstpolarization weight vector including the polarization weights of the Npolarized channels, where the polarization weights are a basis forinformation bit selection; then determines positions of the informationbits based on the first polarization weight vector; and finally performspolar encoding on the N to-be-encoded bits to obtain the polar-encodedbits. Therefore, the position of the information bits and those of thefrozen bits are determined by calculating polarization weights ofpolar-code polarized channels, without relating to a channel parameterand a code rate. This can reduce calculation complexity of polar codeencoding. In addition, polarization weights corresponding to polar codeswith different code lengths and code rates have a same part, so thatstorage space is greatly reduced, and storage complexity of polar codeencoding can also be reduced.

Optionally, a target code length M of a polar code is equal to apositive integer power of 2, and the obtaining module 12 is specificallyconfigured to obtain the first polarization weight vector from aprestored second polarization weight vector. The second polarizationweight vector includes polarization weights of N_(max) polarizedchannels, N_(max) is a preset mother code length corresponding to amaximum code length supported by a communications system, and the mothercode length is a positive integer power of 2.

In this implementation, the polarization weights are obtained in anoffline storage manner. Compared with online calculation, when there isenough storage space, the offline storage manner can shorten latencycaused by online calculation.

Optionally, when the target code length M of the polar code is less thanor equal to N_(max), an element set in the first polarization weightvector is a subset of an element set in the second polarization weightvector; or when the target code length M of the polar code is greaterthan or equal to N_(max), a polarization weight of a polarized channelwhose channel index number is less than or equal to N_(max) is in aone-to-one correspondence with a polarization weight in the secondpolarization weight vector, and a polarization weight of a polarizedchannel whose channel index number is greater than N_(max) is equal to aweight increment

$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$

plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), where ϕ andα are parameters that are preset based on the target code length and acode rate of the polar code.

In this implementation, a half-online-calculation half-offline-storagemanner is used, so that not only latency caused by online calculationcan be shortened, but also storage complexity can be reduced.

In the foregoing implementation, further, ϕ=0, and α=¼.

Further, a target code length M of a polar code is not equal to apositive integer power of 2, and the first determining module 11 isspecifically configured to determine that N=2^(└log) ² ^((M−1)┘+1),where the symbol └ ┘ denotes rounding down.

FIG. 9 is a schematic structural diagram of Embodiment 2 of an encodingdevice according to this application. As shown in FIG. 9, based on astructure of the encoding device shown in FIG. 8, the encoding device200 in this embodiment may further include a third determining module 15and a calculation module 16. The third determining module 15 isconfigured to: before the obtaining module 12 obtains the firstpolarization weight vector including the polarization weights of the Npolarized channels, determine positions of N−M to-be-deleted bits in theN to-be-encoded bits. The calculation module 16 is configured tocalculate, based on a result of the determining performed by the thirddetermining module 15, a set of modification coefficients including thepolarization weights corresponding to the N to-be-encoded bits. Themodification coefficient is used to represent an amount of modificationto each polarization weight resulting from bit deletion. The obtainingmodule 12 is specifically configured to calculate the polarizationweights of the N polarized channels based on the set of modificationcoefficients, to obtain the first polarization weight vector.

Further, the third determining module 15 is specifically configured to:determine a channel index number sequence P₀ ^(N−1)=[0,1, . . . ,N−1] ofthe N polarized channels; and perform the following operations on eachindex number in the channel index number sequence: performing bitreversal on a binary number corresponding to each index number, andconverting, to a decimal number, a binary number obtained after bitreversal, to obtain a new channel index number sequence P₀ ^(N−1)′;sorting index numbers in the new channel index number sequence P₀^(N−1)′; and selecting, in ascending order, N−M index numbers in thechannel index number sequence P₀ ^(N−1)′ as index numbers of theto-be-deleted bits.

Optionally, the set of modification coefficients is

${g = \begin{bmatrix}g_{({0,0})} & {\; \ldots} & \; & g_{({0,{n - 1}})} \\g_{({1,0})} & \; & \; & g_{({1,{n - 1}})} \\\vdots & {\; \vdots} & \; & \vdots \\g_{({{N - 1},0})} & \; & \; & g_{({{N - 1},{n - 1}})} \\g_{({N,0})} & g_{({N,1})} & \ldots & g_{({N,{n - 1}})}\end{bmatrix}};$

and

the calculation module 16 is specifically configured to calculate eachitem g_((i,j)) in the set g of modification coefficients according tothe following formulas:

g _((i,j))=(p _(i) +p _(i+2) _(j−2) )/2; and

g _((i+2) _(j) _(,j)) =g _((i,j)), where

iϵ{0,1, . . . ,N−1}, jϵ{0,1, . . . ,n−1}, N=2^(n), P_(i) is anidentifier indicating whether a bit corresponding to a polarized channelwhose channel index number is i is to be deleted, and P_(i)ϵ{0,1}.

Optionally, the obtaining module 12 is specifically configured tocalculate the polarization weights W_(i) of the N polarized channelsaccording to the following formula, to obtain the first polarizationweight vector:

${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {1 - g_{({i,j})}} )*( {2^{j} + \varphi} )^{\alpha}}}},$

where

i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i. B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), g_((i,j)) is an element in row i and column j inthe set g of modification coefficients, and ϕ and α are parameters thatare preset based on the target code length and a code rate of the polarcode.

Optionally, the obtaining module 12 is specifically configured to obtainthe first polarization weight vector W₀ ^(N−1) through recursioncalculation according to the following formulas:

β_(i)=(2^(i−1)+ϕ)^(α);

W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹ +g _((i,j))*β_(i); and

W ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; where

W ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], where

iϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, g_((i,j)) isan element in row i and column j in the set g of modificationcoefficients, and W₀ is an arbitrary constant.

In the foregoing implementation, ϕ=0, and α=¼.

In the foregoing implementation, the second determining module 13 isspecifically configured to: sort the polarization weights in the firstpolarization weight vector in descending order, and select bitscorresponding to the first K polarization weights as information bits.

The encoding device in this embodiment may be configured to execute thetechnical solution in the method embodiment shown in FIG. 7. Theirimplementation principles are similar, and details are not repeatedherein.

After receiving input information bits, the encoding device provided inthis embodiment determines the quantity N of to-be-encoded bits based onthe target code length M of the polar code; afterwards determines thepositions of the N−M to-be-deleted bits in the N to-be-encoded bits, andcalculates, based on the result of the determining, the set ofmodification coefficients including modification coefficients of the Nto-be-encoded bits; then calculates the polarization weights of the Npolarized channels based on the set of modification coefficients, toobtain the first polarization weight vector; and finally determinespositions of the information bits based on the first polarization weightvector, and performs polar encoding on the N to-be-encoded bits toobtain the polar-encoded bits. Therefore, for a polar code with anytarget code length, positions of information bits and those of frozenbits are determined by calculating polarization weights of polar-codepolarized channels, without relating to a channel parameter and a coderate. This can reduce calculation and implementation complexity of polarcode encoding.

A person of ordinary skill in the art may understand that some or all ofthe steps in the method embodiments may be implemented by a programinstructing relevant hardware. The program may be stored in acomputer-readable storage medium. When the program runs, the steps inthe method embodiments are performed. The storage medium includesvarious media that can store program code, such as a ROM, a RAM, amagnetic disk, or an optical disc.

As shown in the foregoing method and the corresponding device, when thepolarization weights in the first polarization weight vector are sortedin descending order, according to a method for calculating apolarization weight according to a formula or a polynomial correspondingto a binary representation manner of a polarized channel sequencenumber, if a polarization weight of one polarized channel is greaterthan a polarization weight of another polarized channel, a position ofthe “one polarized channel” in the first polarization weight vector isnecessarily before that of the “another polarized channel”. Certainly,when the polarization weights in the first polarization weight vectorare sorted in ascending order, if a polarization weight of one polarizedchannel is greater than a polarization weight of another polarizedchannel, a position of the “one polarized channel” in the firstpolarization weight vector is necessarily after that of the “anotherpolarized channel”. In addition, a position of each polarized channel inthe first polarization weight vector may be considered as a polarizationweight multiplied by a factor.

According to the foregoing method and the corresponding device forgenerating the first polarization weight vector including thepolarization weights of the N polarized channels, the following severalexamples of the first polarization weight vector may be provided. Thepolarization weights may be sorted in descending order, or thepolarization weights may be sorted in ascending order. Without loss ofgenerality, sequence numbers in all the examples are ranked from 1, inother words, a minimum sequence number value is 1. This is equivalent tothat the sequence numbers are ranked from 0. If the sequence numbers areranked from 0, 1 only needs to be subtracted from each sequence numberin a corresponding sequence.

The polarization weights in the first polarization weight vector may besorted in descending order. Herein, a minimum sequence number valueis 1. Correspondingly, if a minimum sequence number value is 0, 1 may besubtracted from each of all sequence numbers.

(1) N=16, and the first polarization weight vector is:

16,15,14,12,8,13,11,10,76,4,9,5,3,2,1.

(2) N=32, and the first polarization weight vector is:

32,31,30,28,24,16,29,27,26,23,22,15,20,14,12,25,8,21,19,13,18,11,10,7,6,4,17,9,5,3,2,1.

(3) N=64, and the first polarization weight vector is:

64,63,62,60,56,48,61,32,59,58,55,54,47,52,46,31,44,30,57,40,28,53,24,51,45,50,16,43,29,42,39,27,38,26,23,36,22,49,15,20,14,41,12,37,25,8,35,21,34,19,13,18,11,10,7,6,33,4, 17,9,5,3,2,1.

(4) N=128, and the first polarization weight vector is:

128,127,126,124,120,112,125,96,123,122,119,64,118,111,116,110,95,108,94,121,104,92,63,117,62,88,115,109,60,114,80,107,56,93,106,103,91,48,102,90,61,87,100,32,59,86,113,79,58,84,55,78,105,54,76,47,101,52,89,46,72,99,31,44,85,98,30,57,83,40,77,28,82,53,75,24,51,74,45,71,50,16,43,70,97,29,42,68,39,27,81,38,26,23,36,73,22,49,15,69,20,14,41,67,12,66,37,25,8,35,21,34,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

(5) N=256, and the first polarization weight vector is:

256,255,254,252,248,240,253,224,251,250,247,192,246,239,244,238,128,223,236,222,249,232,220,191,245,190,216,243,237,188,242,127,208,235,184,126,221,234,231,124,219,176,230,218,189,120,215,228,160,187,214,241,112,207,186,212,183,125,206,233,182,96,123,204,175,229,180,122,217,174,119,200,227,64,159,118,172,213,226,111,158,185,116,211,168,110,205,156,210,181,95,108,203,152,179,94,121,202,173,104,199,178,92,63,144,117,171,198,225,62,157,88,170,115,196,167,109,60,155,114,209,166,80,107,54,56,151,164,93,106,201,103,150,177,91,48,143,102,197,148,90,61,142,87,169,100,195,32,59,86,140,113,194,165,79,58,153,84,55,136,163,78,105,54,149,162,76,47,101,52,147,89,46,141,72,99,146,31,44,85,139,98,193,30,57,138,83,40,135,77,28,82,53,134,161,75,24,51,132,74,45,71,50,145,16,43,70,97,29,42,137,68,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

(6) N=512, and the first polarization weight vector is:

512,511,510,508,504,496,509,480,507,506,503,448,502,495,500,494,384,479,492,478,505,488,476,447,501,256,446,472,499,493,444,498,383,464,491,440,382,477,490,487,380,475,432,486,474,255,445,376,471,484,254,416,443,470,497,368,463,252,442,468,439,381,462,489,248,438,352,379,460,431,485,436,378,473,240,430,375,456,483,253,320,415,374,428,469,482,367,224,414,251,441,372,467,424,366,461,250,412,466,247,437,351,364,459,192,246,408,435,350,377,458,239,429,360,455,244,434,348,238,319,400,373,427,454,481,128,223,318,413,236,344,426,371,452,423,365,222,249,316,411,370,465,232,422,336,363,220,410,191,245,312,407,420,349,362,457,190,359,216,406,243,433,347,237,304,399,358,188,453,242,404,346,127,208,317,398,235,343,425,356,451,184,126,221,288,234,315,342,396,369,450,231,421,335,124,219,314,409,176,340,230,311,392,419,334,361,218,189,120,215,310,405,228,418,332,303,160,357,187,214,241,308,403,345,112,207,302,397,328,186,355,212,402,183,125,206,287,233,300,341,395,354,449,182,96,286,123,204,313,394,175,339,229,296,391,180,333,122,217,284,174,338,119,200,309,390,227,417,331,64,159,118,172,213,280,226,307,388,330,111,301,158,327,185,116,211,306,401,168,110,205,272,299,326,156,353,210,181,95,285,108,203,298,393,324,295,152,179,94,121,202,283,173,337,104,199,294,389,178,92,282,63,144,117,171,198,279,225,292,387,329,62,157,88,170,278,115,196,305,386,167,109,271,60,325,155,114,209,276,166,80,270,107,297,154,323,56,151,164,93,106,201,268,322,103,293,150,177,91,281,48,143,102,197,264,291,148,90,61,142,87,169,277,100,195,29,385,32,59,86,140,113,194,275,165,79,269,58,153,84,274,55,136,163,78,105,267,321,54,149,162,76,266,47,101,263,52,147,89,46,141,72,262,99,289,146,31,44,85,139,98,193,260,30,57,138,83,273,40,135,77,28,82,53,134,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

(7) N=1024, and the first polarization weight vector is:

1024,1023,1022,1020,1016,1008,1021,992,1019,1018,1015,960,1014,1007,1012,1006,896,991,1004,990,1017,1000,988,959,1013,768,958,984,1011,1005,956,1010,895,976,1003,952,894,989,1002,999,512,892,987,944,998,986,767,957,888,983,996,766,928,955,982,1009,880,975,764,954,980,951,893,974,1001,760,950,864,511,891,972,943,997,948,510,890,985,752,942,887,968,995,508,765,832,927,886,940,981,994,879,736,926,763,953,504,884,979,936,878,973,762,924,978,759,949,863,496,876,971,704,758,920,947,862,509,889,970,751,941,872,967,756,946,480,860,507,750,831,912,885,939,966,993,506,640,735,830,925,748,856,938,503,883,964,935,877,734,448,761,828,923,502,882,977,744,934,848,495,875,732,922,703,500,757,824,919,932,861494,874,969,702,871,728,918,755,945,384,479,859,492,749,816,911,870,700,965,754,916,478,858,505,639,720,829,910,747,855,937,488,868,963,696,476,638,733,800,447,746,827,854,908,501,881,962,743,933,847,256,446,636,731,826,921,472,688,852,499,742,823,904,931,846,493,873,730,444,701,498,632,727,822,917,740,930,383,464,844,491,815,672,869,699,726,440,753,820,915,382,477,857,490,624,719,814,909,840,698,487,867,724,914,695,380,475,637,718,799,432,745,812,853,907,486,866,961,694,474,608,255,798,445,635,716,825,376,906,471,687,851,484,741,808,903,692,845,254,416,634,729,796,443,470,686,850,497,631,712,821,902,739,929,368,463,843,252,442,576,671,468,630,684,725,792,439,738,819,900,381,462,842,489,623,813,670,839,697,248,438,628,723,818,913,352,680,379,460,622,717,784,431,811,838,668,485,865,722,436,693,378,473,607,240,797,430,620,75,810,375,905,456,836,483,807,664,691,606,253,320,415,633,714,795,374,428,469,685,849,482,616,711,806,901,690,367,224,414,604,251,794,441,575,656,372,467,629,683,710,791,424,737,804,899,366,461,841,250,412,574,669,466,600,682,247,790,437,627,708,817,898,351,679,364,459,621,783,192,572,837,667,246,408,626,721,788,435,350,678,377,458,592,239,782,429,619,809,360,666,455,835,244,434,568,663,348,676,605,238,319,400,618,713,780,373,427,454,834,481,615,805,662,689,128,223,318,413,603,236,793,344,426,560,655,371,452,614,709,776,423,803,660,365,222,602,249,316,411,573,654,370,465,599,681,232,789,422,612,707,802,897,336,363,220,410,544,191,571,598,652,245,312,407,625,706,787,420,349,677,362,457,591,781,190,570,359,665,216,406,596,243,786,433,567,648,347,675,590,237,304,399,617,779,358,188,453,833,242,404,566,661,346,674,127,208,317,398,588,235,778,343,425,559,356,451,613,775,184,564,659,126,221,288,601,234,315,342,396,558,653,369,450,584,231,774,421,611,801,658,335,124,219,314,409,543,176,340,556,597,651,230,311,392,610,705,772,419,334,361,218,542,189,569,120,650,215,310,405,595,228,785,418,552,647,332,589,303,160,540,357,187,214,594,241,308,403,565,646,345,673,112,207,302,397,587,777,328,186,355,212,402,536,183,563,644,125,206,287,586,233,300,341,395,557,354,449,583,773,182,562,657,96,286,123,204,313,394,528,175,339,555,582,229,296,391,609,771,180,333,122,217,284,541,174,338,554,119,649,200,309,390,580,227,770,417,551,331,64,159,539,118,172,213,280,593,226,307,388,550,645,330,111,301,158,538,327,185,116,211,306,401,535,168,548,643,110,205,272,585,299,326,156,353,210,534,181,561,642,95,285,108,203,298,393,527,324,581,295,152,532,179,94,121,202,283,526,173,337,553,104,199,294,389,579,769,178,92,282,63,144,524,117,171,198,279,578,225,292,387,549,329,62,157,537,88,170,278,115,196,305,386,520,167,547,109,271,60,325,155,114,209,276,533,166,546,641,80,270,107,297,154,323,56,151,531,164,93,106,201,268,525,322,103,293,150,530,177,91,281,48,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,32,59,86,140,113,194,275,518,165,545,79,269,58,153,84,274,55,136,516163,78,105,267,321,54,149,529,162,76,266,47,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

(8) N=2048, and the first polarization weight vector is:

2048,2047,2046,2044,2040,2032,2045,2016,2043,2042,2039,1984,2038,2031,2036,2030,1920,2015,2028,2014,2041,2024,2012,1983,2037,1792,1982,2008,2035,2029,1980,2034,1919,2000,2027,1976,1918,2013,2026,2023,1536,1916,2011,1968,2022,2010,1791,1981,1912,2007,2020,1790,1952,1979,2006,2033,1904,1999,1788,1978,2004,1975,1917,1998,2025,1024,1784,1974,1888,1535,1915,1996,1967,2021,1972,1534,1914,2009,1776,1966,1911,1992,2019,1532,1789,1856,1951,1910,1964,2005,2018,1903,1760,1950,1787,1977,1528,1908,2003,1960,1902,1997,1786,1948,1023,2002,1783,1973,1887,1520,1900,1995,1728,1022,1782,1944,1971,1886,1533,1913,1994,1775,1965,1896,1991,1020,1780,1970,1504,1884,1531,1774,1855,1936,1909,1963,1990,2017,1016,1530,1664,1759,1854,1949,1772,1880,1962,1527,1907,1988,1959,1901,1758,1472,1785,1852,1947,1526,1906,1008,2001,1768,1958,1872,1519,1899,1756,1946,1727,1021,1524,1781,1848,1943,1956,1885,1518,1898,1993,1726,992,1895,1752,1019,1942,1779,1969,1408,1503,1883,1516,1773,1840,1935,1894,1724,1989,1018,1778,1940,1502,1015,1882,1529,1663,1744,1853,1934,1771,1879,1961,1512,1892,1987,960,1720,1014,1500,1662,1757,1824,1471,1770,1851,1878,1932,1525,1905,1007,1986,1767,1957,1871,1012,1280,1470,1660,1755,1850,1945,1496,1712,1876,1006,1523,1766,1847,1928,1955,1870,1517,1897,1754,1468,1725,896,991,1522,1656,1751,1846,1004,1941,1764,1954,1407,14881868,1515,1839,1696,990,1893,1723,1750,1017,1464,1777,1844,1939,1406,1501,1000,1881,1514,1648,1743,1838,1933,1864,1722,1511,988,1891,1748,959,1938,1719,1013,1404,1499,1661,1742,1823,1456,1769,1836,1877,1931,1510,1890,768,1985,958,1718,984,1498,1632,1011,1279,1822,1469,1659,1740,1849,1400,1930,1495,1711,1875,1005,1508,1765,1832,1927,956,1716,1869,1010,1278,1440,1658,1753,1820,1467,1494,1710,895,1874,976,1521,1655,1736,1845,1003,1926,1763,1953,1392,1487,1867,952,1276,1466,1600,1695,894,989,1492,1654,1708,1749,1816,1002,1463,1762,1843,1924,1405,1486,999,1866,1513,1647,1837,1694,512,1863,1721,892,1272,987,1462,1652,1747,1842,944,1937,1376,1704,998,1403,1484,1646,1741,1808,1455,1835,1862,1692,1509,986,1889,767,1746,957,1460,1717,888,983,1402,1497,1631,996,1264,1821,1454,1644,1739,1834,1399,1929,1480,1860,766,1507,928,1831,1688,955,982,1715,1630,1009,1277,1344,1439,1657,1738,1819,1398,1452,1493,1709,880,1873,975,1506,1640,1735,1830,764,1925,954,1714,1391,980,1248,1438,1628,951,1275,1818,1465,1599,680,893,1396,974,1491,1653,1707,1734,1815,1001,1448,1761,1828,1923,1390,1485,760,1865,950,1274,1436,1598,1693,864,511,1490,1624,1706,891,1271,1814,972,1461,1651,1732,1841,943,1922,1375,1703,997,1388,1483,1645,1807,948,1216,510,1596,1861,1691,890,1270,985,1432,1650,752,1745,1812,942,1459,1374,1702,887,968,1401,1482,1616,995,1263,1806,1453,1643,1833,1384,1690,1479,508,1859,765,832,1268,927,1458,1592,1687,886,940,981,1372,1700,1629,994,1262,1343,1424,1642,1737,1804,1397,1451,1478,879,1858,736,1505,1639,926,1829,763,1686,953,504,1713,884,1152,979,1247,1342,1437,1627,936,1260,1817,1368,1450,1584,1679,878,1395,973,1476,1638,1733,1800,762,1447,924,1827,1684,1389,978,1246,759,1626,949,1273,1340,1435,1597,1678,863,1394,496,1489,1623,1705,876,1256,1813,971,1446,1636,1731,1826,704,1921,1360,758,1387,920,1244,1434,1568,947,1215,862,509,1595,1622,1676,889,1269,1336,970,1431,1649,751,1730,1811,941,1444,1373,1701,872,967,1386,1481,1615,756,1805,946,1214,480,1594,1383,1689,860,1240,507,1430,1620,750,8311267,1810,912,1457,1591,1672,885,939,966,1371,1699,1614,993,1261,1328,1423,1641,1803,1382,1212,1477,506,640,1857,735,830,1266,925,1428,1590,748,1685,856,938,503,1370,1698,883,1151,964,1232,1341,1422,1612,935,1259,1802,1367,1449,1583,877,1380,734,1475,1637,448,1799,761,828,1208,923,502,1588,1683,882,1150,977,1245,1312,744,1625,934,1258,1339,1366,1420,1582,1677,848,1393,495,1474,1608,875,1255,1798,732,1445,1635,922,1825,703,1682,1359,500,757,824,1148,919,1243,1338,1433,1567,932,1200,861,1364,494,1580,1621,1675,874,1254,1335,969,1416,1634,1729,1796,702,1443,1358,871,728,1385,918,1242,755,1566,945,1213,384,479,1593,1144,1674,859,1239,1334,492,1429,1619,749,816,1252,1809,911,1442,1576,1671,870,700,965,1356,1613,754,1327,916,1184,478,1564,1381,1211,858,1238,505,639,1618,720,829,1265,1332,910,1427,1589,747,1670,855,937,488,1369,1697,868,1136,963,1231,1326,1421,1611,696,1801,1352,1210,476,638,1379,733,800,1236,447,1426,1560,746,827,1207,854,908,501,1587,1668,881,1149,962,1230,1311,743,1610,933,1257,1324,1365,1419,1581,847,1378,256,1473,1607,446,636,1797,731,826,1206,921,472,1586,688,1681,852,1120,499,1310,742,823,1147,904,1228,1337,1418,1552,931,1199,846,1363,493,1579,1606,873,1253,1320,730,1415,1633,444,1795,701,1204,1357,498,632,727,822,1146,917,1241,1308,740,1565,930,1198,383,1362,464,1578,1143,1673,844,1224,1333,491,1414,1604,815,1251,1794,672,1441,1575,869,699,726,1355,440,753,820,1088,915,1183,382,477,1563,1142,1196,857,1237,1304,490,624,1617,719,814,1250,1331,909,1412,1574,1669,840,698,487,1354,867,1135,724,1325,914,1182,695,1562,1351,1209,380,475,637,1140,718,799,1235,1330,432,1425,1559,745,812,1192,853,907,486,1572,1667,866,1134,961,1229,1296,1609,694,1323,1350,1180,474,608,1377,255,798,1234,445,635,1558,716,825,1205,376,906,471,1585,687,1666,851,1119,484,1309,741,808,1132,903,1227,1322,1417,1551,692,845,1348,254,1605,416,634,1319,729,796,1176,443,470,1556,686,1203,850,1118,497,631,712,821,1145,902,1226,1307,739,1550,929,1197,368,1361,463,1577,1128,843,1223,1318,252,1413,1603,442,576,1793,671,1202,468,630,684,725,792,1116,439,1306,738,819,1087,900,1168,381,462,1548,1141,1195,842,1222,1303,489,623,1602,813,1249,1316,670,1411,1573,839,697,248,1353,438,628,723,818,1086,913,1181,352,680,1561,1112,1194,379,1302,460,622,1139,717,784,1220,1329,431,1410,1544,811,1191,838,668,485,1571,865,1133,722,1295,436,693,1084,1349,1179,378,473,607,138,240,797,1233,1300,430,620,1557,715,810,1190,375,905,456,1570,1665,836,1104,4831,294,807,1131,664,1321,178,691,606,1347,253,320,415,633,1080,714,795,1175,374,428,469,1555,685,1188,849,1117,482,616,711,806,1130,901,1225,1292,1549,690,367,1346,224,1127,414,604,1317,251,794,1174,441,575,1554,656,1201,372,467,629,683,710,791,1115,424,1305737,804,1072,899,167,366,461,1547,1126,841,1221,1288,250,1601,412,574,1315,669,172,466,600,682,247,790,1114,437,627,708,817,1085,898,1166,351,679,1546,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,572,837,667,246,408,626,721,788,1056,435,350,678,1083,1110,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,809,1189,360,666,455,1569,835,1103,244,1293,434,568,663,1082,1177,348,676,605,1108,238,319,1298,400,618,1079,713,780,1160,373,427,454,1540,1187,834,1102,481,615,805,1129,662,1291,689,128,1345,223,318,413,603,1078,236,793,1173,344,426,560,1553,655,1186,371,452,614,709,776,1100,423,1290,803,1071,660,365,222,1125,602,1287,249,316,411,573,1076,654,1171,370,465,599,681,232,789,1113,422,612,707,802,1070,897,1165,336,1545,1096,363,1286,220,1123,410,544,1313,191,1170,571,598,652,245,312,407,625,706,787,1055,420,349,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,1541,570,359,665,216,406,596,243,786,1054,433,567,648,1081,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,833,1101,242,404,566,661,1052,346,674,127,1106,208,317,398,588,1077,235,778,1158,343,425,559,1538,1185,356,451,613,775,1099,184,1289,564,659,126,221,288,601,1048,234,315,342,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,801,1069,658,335,1095,124,1285,219,314,409,543,1074,176,1169,340,556,597,651,230,311,392,610,705,772,1040,419,334,1067,1094,361,218,1121,542,1283,189,569,120,650,215,310,405,595,228,785,1053,418,552,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,594,241,308,403,565,646,1051,345,673,112,1105,207,302,397,587,1062,777,1157,328,186,1537,355,212,402,536,183,1050,563,644,125,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,562,657,96,286,123,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,180,333,1093,122,217,284,541,1044,174,338,554,119,649,200,309,390,580,227,770,1038,417,551,1065,331,1091,64,1281,159,539,118,172,213,280,593,226,307,388,550,645,1036,330,111,1090,301,158,1061,538,327,185,116,211,306,401,535,168,1049,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,561,642,95,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,106,201,268,525,1028,322,103,293,150,530,177,91,281,48,1041,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

(9) N=4096, and the first polarization weight vector is:

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172,2094,466,600,2189,682,247,790,114,2569,2120,437,627,2310,708,817,1085,2147,898,1166,351,2337,679,1546,2194,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,2079,572,837,667,246,2092,2133,408,626,2187,721,788,1056,2146,435,350,2241,2308,678,1083,1110,2565,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,2078,809,1189,360,666,455,1569,2105,835,2186,1103,244,1293,2131,434,568,2321,663,1082,2088,1177,348,2183,676,605,1108,2563,238,319,1298,2125,400,618,1079,713,780,1160,373,427,454,1540,2076,1187,834,1102,481,615,2130,805,1129,662,1291,2101,2182,689,128,1345,2562,223,318,2209,413,603,1078,236,793931173,344,2123,426,560,1553,2313,655,1186,371,3073,452,614,709,776,1100,423,1290,2072,803,1071,660,365,2099,222,2180,1125,602,1287,249,316,2122,411,573,1076654,1171,370,2093,465,599,681,232,789,1113,2119,422,612,2309,707,802,1070,897,1165,336,2098,1545,2193,1096,363,1286,220,1123,410,544,1313,191,1170,2064,571,598,652,245,312,2091,2118,407,625,706,787,1055,2145,420,349,2307,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,1541,2077,570,359,665,216,2090,406,596,2185,243,786,1054,2116,433,567,2306,648,1081,2087,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,2075,833,1101,242,2129,404,566,661,1052,2086,346,2181,674,127,1106,2561,208,317,398,588,1077,235,778,1158,343,425,559,1538,2074,1185,356,451,613,775,1099,184,1289,2071,564,659,126,2084,221,288,2179,601,1048,234,315,342,2121,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,2070,801,1069,658,335,2097,2178,1095,124,1285,219,314,409,543,1074,176,1169,340,2063,556,597,651,230,311,2117,392,610,705,772,1040,419,334,2068,1067,1094,361,218,1121,542,1283,189,2062,569,120,650,215,310,2089,405,595,228,785,1053,2115,418,552,2305,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,2060,594,241,308,2114,403,565,646,1051,2085,345,673,112,1105,207,302,397,587,1062,777,1157,328,186,1537,2073,355,212,402,536,183,1050,2056,563,644,125,2083,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,2069,562,657,96,2082,286,2177,123,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,180,333,2067,1093,122,217,284,541,1044,174,338,2061,554,119,649,200,309,390,580,227,770,1038,417,551,2066,1065,331,1091,64,1281,159,539,118,172,213,280,2059,593,226,307,2113,388,550,645,1036,330,1111090,301,158,1061,538,327,185,2058,116,211,306,401,535,168,1049,2055,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,2054,561,642,95,2081,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,2052,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,2065,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,2057,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,2053,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,2051,106,201,268,525,1028,322,103,293,150,530,177,2050,91,281,48,1041,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,2049,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.

The polarization weights in the first polarization weight vector mayalternatively be sorted in ascending order. Herein, a minimum sequencenumber value is 1. Similarly, if a minimum sequence number value is 0,only subtracting 1 from all sequence numbers is required.

(1) N=16, and the first polarization weight vector is:

1,2,3,5,9,4,6,7,10,11,13,8,12,14,15,16.

(2) N=32, and the first polarization weight vector is:

1,2,3,5,9,17,4,6,7,10,11,18,13,19,21,8,25,12,14,20,15,22,23,26,27,29,16,24,28,30,31,32.

(3) N=64, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,11,18,13,19,34,21,35,8,25,37,12,41,14,20,15,49,22,36,23,26,38,27,39,42,29,43,16,50,45,51,24,53,28,40,57,30,44,31,46,52,47,54,55,58,59,32,61,48,56,60,62,63,64.

(4) N=128, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,34,21,35,8,25,37,66,12,67,41,14,20,69,15,49,22,73,36,23,26,38,81,27,39,68,42,29,97,70,43,16,50,71,45,74,51,24,75,53,82,28,77,40,83,57,30,98,85,44,31,99,72,46,89,52,101,47,76,54,105,78,55,84,58,79,113,86,59,32,100,87,61,90,102,48,91,103,106,93,56,107,80114,60,109,11588,62,117,6392104,121,94,108,95,110,116,111,118,64,119,122,123,96,125,112,120,124,126,127,128.

(5) N=256, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11, 18,13,19,129,34,21,35,8,25,37,66,12,67,41,14, 2069,15,130,49,22,73,131,36,23,26,133,38,81,27,39,68,137,42,29,97,70,43,16,145,50,71,45,74,132,51,24,75,161,134,53,82,28,77,135,40,83,138,57,30,193,98,139,85,44,31,146,99,72,141,46,89,147,52,101,47,76,162,149,54,105,78,163,136,55,84,153,58,79,165,194,113,140,86,59,32,195,100,169,87,142,61,90,148,197,102,143,48,91,177,150,103,201,106,93,164,151,56,154,107,80,166,209,114,155,60,109,167,196,115,170,88,157,62,225,198,171,117,144,63,92178,199,104,173,202,121,94,179,152,203,108,95,181,210,156,205,110,168,211,116,185,158,111,226,213,172,118,159,64,227,200,11119,174,123,96,182,233,206,125,183,212,186,207,112,241,214,187,160,228,215,120,189,218,230,176,219,124,231,234,221,126,184,235,208,127,242,188,237,243,216,190,245,191,220,232,249,222,236,223,128,238,244,239,246,192,247,250,251,224,253,240,248,252,254,255,256.

(6) N=512, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,257,67,41,14,20,69,15,130,49,22,73,131,36,23,26,133,38,81,27,39,258,68,137,42,29,259,97,70,43,16,145,50,261,71,45,74,132,51,24,265,75,161,134,53,82,28,77,135,40,273,83,138,57,30,260,193,98,139,85,44,31,146,289,99,262,72,141,46,89,147,52,263,101,47,266,76,162,149,54,321,267,105,78,163,136,55,274,84,153,58,269,79,165,275,194,113,140,86,59,32,385,290,195,100,277,169,87,142,61,90,148,291,264,197,102,143,48,281,91,177,150,293,103,322,268,201,106,93,164,151,56,323,154,297,107,270,80,166,276,209,114,155,325,60,271,109,167,386,305,196,115,278,170,88,157,62,329,387,292,225,279,198,171,117,144,63,282,92,178,389,294,199,104,337,173,283,202,121,94,179,152,295,324,393,298,203,108,285,95,181,210,353,156,326,299,272,205,110,168,401,306,211,116,185,327,158,301,111,330,388,307,226,280,213,172,118,159,64,331,417,227,390,309,200,119,338,174,284,217,122,333,180,391,296,229,339,175,394,313,204,123,286,96,182,449,354,395,341,300,233,287,206,125,183,402,212,355,186,328,397,302,207,112,345,403,308,241,214,187,357,160,303,332,418,228,405,310,215,120,189,218,361,334,419,392,311,230,340,176,409,314,219,124,335,421,231,450,369,396,342,315,234,288,221,126,184,451,356,425,343,235,398,317,208,127,346,404,242,453,188,358,399,304,237,347,433,243,406,216,359,190,457,362,349,420,407,312,245,191,410,220,363,336,422,232,465,370,411,316,249,222,365,423,452,371,426,344,236,413,318,223,128,481,454,427,373,400,319,238,348,434,244,455,360,429,239,458,377,350,435,408,246,192,459,364,351,437,247,466,412,250,461,366,424,467,372,441,251,414,224,367,482,469,428,374,415,320,253,483,456,375,430,240,473,378,436,485,431,460,379,352,438,248,489,462,381,439,468,442,252,463,368,497,470,443,416,254,484,471,376,445,255,474,486,432,475,380,487,490,477,382,440,491,464,383,498,444,493,499,472,446,256,501,447,476,488,505,478,492,479,384,494,500,495,502,448,503,506,507,480,509,496,504,508,510,511,512.

(7) N=1024, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,257,67,41,14,20,69,15,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,70,43,16,145,50,261,71,45,74,132,51,24,265,75,161,514,134,53,8228,77,515,135,40,273,83,138,57,30,517,260,193,98,139,85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,47,266,76,162,529,149,54,321,267,105,78,163,516,136,55,274,84,153,58,269,79,545,165,518,275,194,113,140,86,59,32,519,385,290,195,100,277,169,87,522,142,61,90,148,291,577,264,197,102,523,143,48,281,91,177,530,150,293,103,322,525,268,201,106,93,164,531,151,56,323,154,297,107,270,80,641,546,166,533,276,209,4,155,325,60,209,114,155,32560,271,109,547,167,520,386,305,196,115,278,170,88,537,157,62,329,549,387,292,225,578,279,198,171,117,524,144,63,282,92,178,769,579,389,294,199,104,553,337,173,526,283,202,121,94,179,532,152,295,581,324,527,393,298,203,108,285,95,642,561,181,534,210,353,156,326,299,585,272,205,110,643,548,168,535,401,306,211,116,185,327,538,158,301,111,330,645,550,388,307,226,593,280,213,172,118,539,159,64,331,551,417,770,227,580,390,309,200,649,119,554,338,174,541,284,217,122,333,180,771,609,391,296,229,582,555,339,175,528,394,313,204,123,286,96,657,562,182,773,583,449,354,557,395,341,300,233,586,287,206,125,644,563,183,536,402,212,355,186,328,777,587,397,302,207,112,673,345,646,565,403,308,241,594,214,187,357,540,160,303,589,332,647,552,418,785,228,595,405,310,215,650,120,569,189,542,218,361,334,419,772,705,610,392,311,230,651,597,556,340,176,543,409,314,219,124,335,658,801,611,421,774,231,584,450,369,653,558,396,342,315,234,601,288,221,126,659,564,184,775,613,451,356,559,425,343,778,235,588,398,317,208,127,674,346,661,566,404,242,833,453,188,358,779,617,399,304,237,590,675,347,648,567,433,786,243,596,406,216,665,359,570,190,781,591,457,362,677,349,420,787,706,625,407,312,245,652,598,571,191,544,410,220,363,336,897,802,707,612,422,789,232,681,599,465,370,654,573,411,316,249,602,222,365,660,803,423,776,709,614,452,371,655,560,426,344,793,236,603,413,318,223,128,689,662,805,615,481,834,454,427,373,780,713,618,400,319,238,605,676,348,663,568,434,244,835,455,666,360,809,619,429,782,239,592,458,377,678,350,435,788,721,626,408,246,667,837,572,192,783,621,459,364,679,351,898,817,708,627,437,790,247,682,600,466,669,574,412,250,841,461,366,899,804,737,424,791,710,683,629,467,372,656,575,441,794,251,604,414,224,367,690,901,806,711,616,482,849,685,469,428,374,795,714,633,415,320,253,606,691,664,807,483,836,456,905,375,810,715,620,430,797,240,607,473,378,693,436,722,865,485,668,838,811,431,784,717,622,460,379,680,352,913,818,723,628,438,248,697,839,670,813,623,489,842,462,381,900,819,738,439,792,725,684,630,468,671,576,442,252,843,463,368,929,739,902,821,712,631,497,850,686,470,443,796,729,634,416,254,845,692,903,808,741,484,851,687,471,906,376,825,716,635,445,798,255,608,474,694,961,866,486,907,853,812,745,432,799,718,637,475,380,695,914,724,867,487,698,840,909814,719,624,490,857,477,382,915,820,753,440,726,699,869,672,815,491,844,464,383,930,740,917,822,727,632,498,701,444,730,873,493,846,931,904,823,742,499,852,688,472,921,826,731,636,446,256,847,933,743,962,881,501,908,854,827,746,447,800,733,638,476,696,963,868,488,937,855,747,910,829,720,639,505,858,478,916,754,965,700,870,911,816,749,492,859,479,384,945,755,918,728,871,702,969,874,494,861,932,919,824,757,500,703,922,732,875,495,848,934,744,977,882,502,923,828,761,448,734,877,935,964,883,503,938,856,748,925,830,735,640,506,993,966,939,885,912,831,750,507,860,480,946,756,967,872,941,751,970,889,509,862,947,920,758,704,971,876,496,863,949,759,978,924,762,973,878936,979,884,504,953,763,926,736,879,994,981,940,886,927,832,765,508,995,968,887,942,752,985,890,510,948,997,943,972,891,511,864,950,760,1001,974,893,951,980,954,764,975,880,1009,982,955,928,766,996,983,888,957,767,986,998,944,987,892,512,999,1002,98894,952,1003,976,895,1010,956,1005,1011,984,958,768,1013,959,988,1000,1017,990,1004,991,896,1006,1012,1007,1014,960,1015,1018,1019,992,1021,1008,1016,1020,1022,1023,1024.

(8) N=2048, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,257,67,41,14,20,69,15,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,70,43,16,145,50,261,71,45,74,132,51,24,1025,265,75,161,514,134,53,82,28,77,515,135,40,273,83,138,57,30,517,260,193,98,139,85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,1026,47,266,76,162,529,149,54,321,1027,267,105,78,163,516,136,55,274,84,153,58,1029,269,79,545,165,518,275,194,113,140,86,59,32,1033,519,385,290,195,100,277,169,87,522,142,61,90,148,291,577,264,197,102,523,143,1041,48,281,91,177,530,150,293,103,322,1028,525,268,201,106,93,164,531,151,56,323,1057,154,297,1030,107,270,80,641,546,166,533,276,209,114,155,325,60,1031,271,109,547,1034,167,520,386,305,196,115,278,170,88,537,157,62,1089,329,1035,549,387,292,225,578,279,198,171,117,524,144,1042,63,282,92,178,1037,769,579,389,294,199,104,553,337,173,1043,526,283,202,121,94,179,532,152,295,581,324,1153,1058,527,393,298,203,1045,108,285,95,642,561,181,534,210,3531059,156,326,299,1032,585,272,205,110,643,548,1049,168,535,401,306,2,1116,185,327,538,1061,158,301,1090,111,330,1036,645,550,388,307,226,593,280,213,172,118,539,159,1281,64,1091,331,1065,551,417,1038,770,227,580,390,309,200,649,119,554,338,174,1044,541,284,217,122,1093,333,180,1039,771,609,391,296,229,582,555,339,1154,175,1073,528,394,313,204,1046,123,286,96,657,562,182,1097,773,583,449,354,1155,1060,557,395,341,300,233,1047,586,287,206,125,644,563,1050,183,536,402,212,355,1537,186,32815757777,1062,587,397,302,207,1105,112,673,345,1051,646,565,403,308,241,594,214,187,357,540,1063,160,1282,303,1092,589,332,1161,1066,647,552,418,1053,785,228,595,405,310,215,650,120,569,189,1283,542,1121,218,361,1094,1067,334,419,1040,772,705,610,392,311,230,651,597,556,340,1169,176,1074,543,409,314,219,1285,124,1095,335,658,1069,801,611,421,1098,774,231,584,450,369,1156,653,1075,558,396,342,315,234,1048,601,288,221,126,659,564,1289,184,1099,775,613,451,356,1185,1538,559,425,343,1158,778,235,1077,588,398,317,208,1106,127,674,346,1052,661,566,404,242,1101,833,1539,453,188,358,1159,779,1064,617,399,1297,304,237,1107,590,675,347,1162,1081,648,567,433,1054,786,243,596,406,216,665,359,570,1541,190,1284,1217,781,1122,591,457,362,1163,1109,1068,677,349,420,1055,787,706,625,407,312,245,652,598,571,1170,191,1313,544,410,1123,220,1286,363,1096,1109545,336,1165,897,1070,802,707,612,422,1113,789,232,681599,465,370,1171,654,1076,573,411,316,249,1287,602,1125,222,365,660,1071,803,1290,423,1100,776,709,614,452,371,1186,655,1553,560,426,344,1173,793,236,1078,603,413,318,223,1345,128,689,1291,662,1129,805,615,481,1102,834,1187,1540,454,427,373,1160,780,713,1079,618,400,1298,319,238,1108,605,676,348,177,1082,663,568,434,1293,244,1103,835,1569,455,666,360,1189,809,1542,619,4291299,1218,782,239,1137,592,458,377,1164,1110,1083,678,350,435,1056,788,721,626,408,246,667,837,572,1543,1409,192,1314,1219,783,1124,621,459,1301,364,1193,1111,1546,679,351,1166,898,1085,817,708,627,437,114,790,247,682,600,466,1172,669,1315,574,412,1601,250,1288,1221,841,1126,1547,461,366,1167,899,1072,804,737,1305,424,1115,791,710,683,629,467,372,1201,656,1554,575,441,1174,794,251,1317,604,414,1127,224,1346,367,690,1549,1292,1225,901,1130,806,711,616,482,1117,849,1188,685,1555,469,428,374,1175,795,714,1080,633,415,320,253,1347,606,691,1178,1321,664,1131,807,1294,483,1104,836,1665,1570,456,905,375,1190,810,715,1557,620,430,1300,1233,797,240,1138,607,473,378,1179,1349,1084,693,436,1295,722,1133,865,1571,485,668,838,1191,811,1544,1410,431,1329,1220,784,717,1139,622,460,1302,379,1194,1112,1561,680,352,1181,913,1086,818,723,628,438,1353,248,697,839,1573,1411,670,1316,1249,813,1602,623,489,1303,1222,842,1195,1141,1548,462,381,1168,900,1087,819,738,1306,439,1116,792,725,684,630,468,1202,671,1793,576,442,1603,1413,252,1318,1223,843,1128,1577,463,1361,368,1197,929,1550,739,1307,1226,902,1145,821,712,631,4971118,850,1203,686,1556,470,443,1176,796,729,1319,634,416,1605,254,1348,845,692,1551,1417,1322,1227,903,1132808,741,1309484,1119,851,1666,687,1585,471,906,376,1205,825,716,1558,635,445,1234,798,255,1377,608,474,1180,1350,1323,694,1609,1296,1229,961,1134,866,1667,1572,486,907,853,1192,812,745,1559,1425,432,1330,1235,799,718,1140,637,475,380,1209,1351,1562,695,1182,914,1325,724,1135,867,1354,487,698,840,1669,1574,1412,909,1331,1250,814,719,1617,624,490,1304,1237,857,1196,1142,1563,477,382,1183,915,1088,820,753,440,1355,726,699,869,1575,1441,672,1794,1251,815,1604,1414,491,1333,1224,844,1673,1143,1578,464,1362,383,1198,930,1565,740,1308,1241,917,1146,822,727,632,498,1357,1204,701,1795,444,1633,1415,730,1320,1253,873,1606,1579,493,1363,846,1199,931,1552,1418,1337,1228,904,1147,823,742,1310,499,1120,852,1681,688,1586,472,921,1206,826,731,1797,636,446,1607,1473,256,1378,847,1581,1419,1365,1324,1257,933,1610,743,1311,1230,962,1149,881,1668,1587,501,908,854,1207,827,746,1560,1426,447,1236,800,733,1379,638,476,1210,1352,1801,696,1611,1421,1326,1231,963,1136,868,1697,1369,488,937,855,1670,747,1589,1427,910,1332,1265,829,720,1618,639,505,1238,858,1211,1381,1564,478,1186,1327,754,1613,1356,965,700,870,1671,1576,1442,911,1809,1252,816,749,1619,1429,492,1334,1239,8591674,1144,1593,479,384,1213,945,1566,755,1242,918,1385,728,871,1358,1443,702,1796,1729,1634,1416,969,1335,1254,874,1675,1621,1580,494,1364,86,11200,932,1567,1433,1338,1243,919,148,824,757,500,1359,1682,703,1825,922,1635,1445,732,1798,1255,875,1608,1474,495,1393,848,1677,1582,1420,1366,1339,1258,934,1625,744,1312,1245,977,1150,882,1683,1588,502,923,1208,828,761,1799,448,1637,1475,734,1380,877,1583,1449,1367,1802,1259,935,1612,1422,1341,1232,964,1151,883,1698,1370,503,938,856,1685,748,1590,1428,925,1266,830,735,1857,640,506,1477,1212,1382,1803,1641,1423,1328,1261,993,1614,1699,1371,966,939,885,1672,1591,1457,912,1810,1267,831,750,1620,1430,507,1240,860,1689,1383,1594,480,1214,946,1805,756,1615,1481,1386,967,872,1701,1373,1444,941,1811,1730,751,1649,1431,970,1336,1269,889,1676,1622,1595,509,862,1215,947,1568,1434,1244,920,1387,758,1360,1921,704,1826,1731,1636,1446,971,1813,1256,876,1705,1623,1489,496,1394,863,1678,1597,1435,1340,1273,949,1626,759,1246,978,1389,1684,1827,924,1447,762,1800,1733,1638,1476,973,1395,878,1679,1584,1450,1368,1817,1260,936,1627,1437,1342,1247,979,1152,884,1713,504,953,1686,763,1829,926,1639,1505,736,1858,879,1478,1451,1397,1804,1737,1642,1424,1343,1262,994,1629,1700,1372,981,940,886,1687,1592,1458,927,1268,832,765,1859,508,1479,1690,1384,1833,1643,1453,1806,1263,995,1616,1482,1401,968,887,1702,1374,1459,942,1812,1745,752,1650,1432,985,1270,890,1691,1861,1596,510,1216,948,1807,1645,1483,1388,997,1703,1375,1922,943,1841,1732,1651,1461,972,1814,1271,891,1706,1624,1490,511,864,1693,1598,1436,1274,950,1865,760,1485,1390,1923,1828,1761,18,1761,1448001,1815173,11707,1653,1491,974,1396,893,1680,1599,1465,1818,1275,951,1628,1438,1248,980,1391,1714,954,1925,764,1830,1735,1640,1506,975,1873,880,1709,1493,1452,1398,1819,1738,1657,1439,1344,1277,1009,1630,1715,982,955,1688,1831,928,1507,766,1860,1480,1929,1399,1834,1739,1644,1454,1821,1264,996,1631,1497,1402,983,888,1717,1460,957,1746,767,1889,986,1509,1692,1862,1835,1455,1808,1741,1646,1484,1403,998,1704,1376,1937,944,1842,1747,1652,1462,987,1272,892,1721,1863,512,1694,1837,1647,1513,1866,999,1486,1405,1924,1843,1762,1463,1002,1816,1749,1708,1654,1492,989,894,1695,1600,1466,1276,952,1867,1487,1392,1953,1763,1926,1003,1845,1736,1655,1521,976,1874,895,1710,1494,1467,1820,1753,1658,1440,1278,1010,1869,1716,956,1927,1832,1765,1508,1005,1875,1711,1495,1930,1400,1849,1740,1659,1469,1822,1279,1011,1632,1498,984,1718,958,1985,768,1890,1510,1931,1877,1836,1769,1456,1823,1742,1661,1499,1404,1013,1719,1938,959,1748,1891,988,1511,1722,1864,1933,1838,1743,1648,15141881,1000,1501,1406,1939,1844,1777,1464,1017,1750,1723,1893,990,1696,1839,1515,1868,1488,1407,1954,1764,1941,1004,1846,1751,1656,1522,991,896,1725,1468,1754,1897,1517,1870,1955,1928,1847,1766,1523,1006,1876,1712,1496,1945,1850,1755,1660,1470,1280,1012,1871,1957,1767,1986,1007,1905,1525,1932,1878,1851,1770,1471,1824,1757,1662,1500,1014,1720,960,1987,1892,1512,1961,1879,1771,1934,1853,1744,1663,1529,1882,1015,1502,1940,1778,1018,1989,1724,1894,1935,1840,1773,1516,1883,1503,1408,1969,1779,1942,1019,1752,1895,992,1726,1993,1898,1518,1885,1956,1943,1848,1781,1524,1021,1727,1946,1756,1899,1519,1872,1958,1768,2001,1008,1906,1526,1947,1852,1785,1472,1758,1901,1959,1988,1907,1527,1962,1880,1772,1949,1854,1759,1664,1530,1016,2017,1990,1963,1909,1936,1855,1774,1531,1884,1504,1970,1780,1020,1991,1896,1965,1775,1994,1913,1533,1886,1971,1944,1782,1022,1728,1995,1900,1520,1887,1973,1783,2002,1023,1948,1786,1997,1902,1960,2003,1908,1528,1977,1787,1950,1760,1903,2018,2005,1964,1910,1951,1856,1789,1532,2019,1992,1911,1966,1776,2009,1914,1534,1972,2021,1967,1996,1915,1535,1888,1974,1784,1024,2025,1998,1917,1975,2004,1978,1788,1999,1904,2033,2006,1979,1952,1790,2020,2007,1912,1981,1791,2010,2022,1968,2011,1916,1536,2023,2026,2013,1918,1976,2027,2000,1919,2034,1980,2029,2035,2008,1982,1792,2037,1983,2012,2024,2041,2014,2028,2015,1920,2030,2036,2031,2038,1984,2039,2042,2043,2016,2045,2032,2040,2044,2046,2047,2048.

(9) N=4096, and the first polarization weight vector is:

1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,3421,35,825,37,66,12,257,67,41,14,20,69,15,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,70,43,16,145,50,261,71,45,74,132,51,24,1025,265,75,161,514,134,53,82,28,77,515,135,40,273,83,138,57,30,517,260,193,98,139,85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,1026,47,266,76,2049,162,529,149,54,321,1027,267,105,78,163,516,136,55,274,84,153,58,1029,269,79,545,165,518,275,194,113,140,86,59,32,1033,519,385,290,195,100,277,169,87,522,142,61,90,148,291,577,264,197,102,523,143,1041,48,281,91,2050,177,530,150,293,103,322,1028,525,268,201,106,2051,93,164,531,151,56,323,1057,154,297,1030,107,270,80,641,546,2053,166,533,276,209,114,155,325,60,1031,271,109,547,1034,167,520,386,305,196,115,2057,278,170,88,537,157,62,1089,329,1035,549,387,292,225,578,279,198,171,117,524,144,1042,63,282,92,2065,178,1037,769,579,389,294,199,104,553,337,173,1043,526,283,202,121,2052,94,179,532,152,295,581,324,1153,1058,527,393,298,203,1045,108,285,2081,95,642,561,2054,181,534,210,353,1059,156,326,299,1032,585,272,205,110,643,548,2055,1049,168,535,401,306,211,116,2058,185,327,538,1061,158,301,1090,111,330,1036,645,550,388,2113,307,226,593,2059,280,213,172,118,539,159,1281,64,1091,331,1065,2066,551,417,1038,770,227,580,390,309,200,649,119,554,2061,338,174,1044,541,284,217,122,1093,2067,333,180,1039,771,609,391,296,229,582,555,339,1154,175,1073,528,394,313,204,1046,123,2177,286,2082,96,657,562,2069,182,1097,773,583,449,354,1155,1060,557,395,341,300,233,1047,586,287,206,2083,125,644,563,2056,1050,183,536,402,212,355,2073,1537,186,328,1157,777,1062,587,397,302,207,1105,112,673,345,2085,1051,646,565,403,2114,308,241,594,2060,214,187,357,540,1063,160,1282,303,1092,589,332,1161,1066,647,2305,552,418,2115,1053,785,228,595,405,2089,310,215,650,120,569,2062,189,1283,542,1121,218,361,1094,1067,2068,334,419,1040,772,705,610,392,2117,311,230,651,597,556,2063,340,1169,176,1074,543,409,314,219,1285,124,1095,2178,2097,335,658,1069,801,2070,611,421,1098,774,231,584,450,369,1156,653,1075,558,396,2121,342,315,234,1048,601,2179,288,221,2084,126,659,564,2071,1289,184,1099,775,613,451,356,1185,2074,1538,559,425,343,1158,778,235,1077,588398,317,208,2561,1106,127,674,2181,346,2086,1052,661,566,404,2129,242,1101,833,2075,1539,453,188,358,1159,779,1064,617,399,1297,304,237,1107,590,675,347,1162,2087,1081,648,2306,567,433,2116,1054,786,243,2185,596,406,2090,216,665,359,570,2077,1541,190,1284,1217,781,1122,591,457,362,1163,1109,1068,677,2307,349,420,2145,1055,787,706,625,407,2118,2091,312,245,652,598,571,2064,1170,191,1313,544,410,1123,220,1286,363,1096,2193,1545,2098,336,1165,897,1070,802,707,2309,612,422,2119,1113,789,232,681,599,465,2093,370,1171,654,1076,573,411,2122,316,249,1287,602,1125,2180,222,2099,365,660,1071,803,2072,1290,423,1100,776,709,614,452,3073,371,1186,655,2313,1553,560,426,2123,344,1173,793,236,1078,603,413,2209,318,223,2562,1345,128,689,2182,2101,1291,662,1129,805,2130,615,481,1102,834,1187,2076,1540,454,427,373,1160,780,713,1079,618,400,2125,1298,319,238,2563,1108,605,676,2183,348,1177,2088,1082,663,2321,568,434,2131,1293,244,1103,2186,835,2105,1569,455,666,360,1189,809,2078,1542,619,429,1299,1218,782,239,1137,592,458,377,1164,2565,1110,1083,678,2308,2241,350,435,2146,1056,788,721,2187,626,408,2133,2092,246,667,837,572,2079,1543,1409,192,1314,1219,783,1124,621,459,1301,364,1193,1111,2194,1546,679,2337,351,1166,898,2147,1085,817,708,2310,627,437,2120,2569,1114,790,247,682,2189,600,466,2094,1172,669,1315,574,412,2137,1601,250,1288,1221,841,1126,2195,1547,461,2100,366,1167,899,1072,804,737,2311,1305,424,2149,115,791,710,683,629,467,3074,2095,372,1201,656,2314,1554,575,441,2124,1174,794,251,1317,604,414,1127,2210,224,2577,1346,367,690,2197,1549,2102,1292,1225,901,1130,806,711,2369,616,482,3075,1117,849,1188,685,2315,1555,469,428,2153,374,1175,795,714,1080,633,415,2126,2211,320,253,2564,1347,606,691,2184,1178,2103,1321,664,2322,1131,807,2132,1294,483,1104,2201,836,1665,2106,1570,456,3077,905,375,1190,810,715,2317,1557,620,430,2127,1300,1233,797,240,2593,1138,607,473,2213,378,1179,2566,1349,1084,693,2323,2242,436,2161,1295,722,1133,2188,865,2134,2107,1571,485,668,838,1191,811,2080,1544,1410,431,1329,1220,784,717,1139,622,460,3081,1302,379,1194,2567,1112,2433,1561,680,2338,2243,352,1181,913,2148,1086,818,723,2325,2217,628,438,2135,2570,1353,248,697,2190,839,2109,1573,1411,670,1316,1249,813,2138,1602,623,489,1303,1222,842,1195,1141,2196,1548,462,2339,381,1168,900,2625,1087,819,738,2312,2245,1306,439,2150,2571,1116,792,725,684,2191,630,468,3089,2096,1202671,2329,1793,576,442,2139,1603,1413,252,1318,1223,843,1128,2225,1577,463,2578,1361,368,1197,929,2198,1550,739,2341,1307,1226,902,21511145821,712,2370,631,497,3076,2573,1118,850,1203,686,2316,2249,1556,470,443,2154,1176,796,729,1319,634,4162141,1605,2212,254,2579,1348,845,692,2199,1551,1417,2104,1322,1227,903,1132,808,741,2371,1309,484,3105,1119,2202,851,1666,687,2345,1585,471,3078,906,2155,376,1205,825,716,2318,1558,635,445,2128,268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02,1023,3630,3468,3414,3387,3306,2982,1948,3673,2792,1786,3360,3293,3025,3198,2930,3731,1997,3636,2550,1902,2971,3256,2876,2809,3847,2496,1960,3685,3523,2782,3428,2925,2003,3631,3497,1908,3415,3850,3307,2983,1528,1977,3660,3470,1787,3389,3280,3012,1950,3199,2931,3746,1760,3418,2551,1903,2986,2904,3733,2796,3638,3476,2973,3314,2878,2783,3905,2688,2554,2018,3525,3260,3430,3851,2005,1964,3689,3471,1910,3376,3309,3041,1951,3662,3747,1856,1789,3419,3014,2987,1532,2933,3720,3639,3505,2960,3858,3315,2879,2798,3668,3478,2555,2019,3288,2908,3737,3431,3642,2528,1992,3262,1911,2994,3853,2804,1966,3663,3529,1776,3434,3015,2009,2920,3749,1914,3421,3492,2989,1534,3859,3778,2799,1972,3697,3479,3018,3384,3317,2937,3724,3670,3643,2557,2021,2910,3263,2995,3616,3482,1967,3292,2968,3435,2806,1996,1915,3408,3969,2752,1535,3874,3779,1888,3684,3494,3019,3861,3304,2924,3753,1974,3671,3537,2544,1784,3442,2911,3726,1024,3645,3483,3388,3321,2997,3674,2807,2025,3294,3026,3437,3732,1998,3875,1917,2972,3495,2810,3848,3781,1975,3686,3524,3021,3443,2926,3727,2004,3632,3498,3416,3865,3308,2984,1978,3675,3485,1788,3390,3295,3027,3200,2932,3761,1999,2552,1904,3001,3734,2811,3877,2974,3687,3553,2784,3906,2927,2033,3526,34993445,3852,3785,2006,1979,3690,3472,3391,3310,3042,1952,3677,3748,1790,3420,3029,2988,2934,3735,3640,3506,2975,3316,2880,2813,3907,2556,2020,3527,3738,3432,3881,2007,3691,3501,1912,3854,3311,3043,1981,3664,3530,1791,3449,3016,2010,2935,3750,3422,3507,2990,3860,3793,2800,3698,3480,3033,3318,2938,3739,3909,3644,2558,2022,3264,2996,3855,1968,3693,3531,3436,3045,2011,3751,1916,3423,3970,2991,1536,3889,3780,3699,3509,3020,3862,3319,2939,3754,3672,3538,2559,2023,2912,3741,3646,3484,3322,2998,3913,2808,2026,3533,3438,3971,2013,3876,3809,1918,3496,3049,3863,3782,3755,1976,3701,3539,3022,3444,2941,3728,3647,3513,3866,3323,2999,3676,3486,2027,3296,3028,3439,3762,2000,1919,3002,3973,2812,3878,3783,3688,3554,3023,3921,2928,3757,2034,3541,3500,3446,3867,3786,1980,3705,3487,3392,3325,3057,3678,3763,2029,3030,3003,3736,3879,2976,3555,2814,3908,2035,3528,3977,3447,3882,3787,2008,3692,3502,3869,3312,3044,1982,3679,3545,1792,3450,3031,2936,3765,3508,3005,3794,2815,3937,3034,3557,3740,3910,3883,2037,3503,3856,3789,1983,3694,3532,3451,3046,2012,3752,3424,3985,2992,3890,3795,3700,3510,3035,3320,2940,3769,3911,2560,2024,3742,3885,3695,3561,3914,3047,2041,3534,3453,3972,2014,3891,3810,3511,3050,3864,3797,3756,3702,3540,3037,2942,3743,3648,3514,3324,3000,3915,2028,3535,3440,4001,2015,3811,1920,3974,3051,3893,3784,3703,3569,3024,3922,2943,3758,3542,3515,3868,3801,3706,3488,3326,3058,3917,3764,2030,3004,3975,3880,3813,3556,3053,3923,3759,2036,3543,3978,3448,3897,3788,3707,3517,3870,3327,3059,3680,3546,2031,3032,3766,3006,4033,2816,3938,3558,3979,3925,3884,3817,2038,3504,3871,3790,1984,3709,3547,3452,3061,3767,3986,3007,3796,3939,3036,3559,3770,3912,2039,3981,3886,3791,3696,3562,3929,3048,2042,3549,3454,3987,3892,3825,3512,3065,3798,3771,3941,3038,3744,3887,3563,3916,2043,3536,3455,4002,2016,3812,3989,3052,3894,3799,3704,3570,3039,2944,3773,3516,3802,3945,3565,3918,4003,2045,3976,3895,3814,3571,3054,3924,3760,3544,3993,3898,3803,3708,3518,3328,3060,3919,2032,4005,3815,4034,3055,3953,3573,3980,3926,3899,3818,3519,3872,3805,3710,3548,3062,3768,3008,4035,3940,3560,4009,3927,3819,2040,3982,3901,3792,3711,3577,3930,3063,3550,3988,3826,3066,4037,3772,3942,3983,3888,3821,3564,3931,2044,3551,3456,4017,3827,3990,3067,3800,3943,3040,3774,4041,3946,3566,3933,4004,2046,3991,3896,3829,3572,3069,3775,3994,3804,3947,3567,3920,2047,4006,3816,4049,3056,3954,3574,3995,3900,3833,3520,3806,3949,4007,4036,3955,3575,4010,3928,3820,3997,3902,3807,3712,3578,3064,4065,40384011,3957,3984,3903,3822,3579,3932,3552,4018,3828,30684039,3944,4013,3823,4042,3961,3581,3934,4019,3992,3830,3070,3776,4043,3948,3568,3935,2048,4021,3831,4050,3071,3996,3834,4045,3950,4008,4051,3956,3576,4025,3835,3998,3808,3951,4066,4053,4012,3958,3999,3904,3837,3580,4067,4040,3959,4014,3824,4057,3962,3582,4020,4069,4015,4044,3963,3583,3936,4022,3832,3072,4073,4046,3965,4023,4052,4026,3836,4047,3952,4081,4054,4027,4000,3838,4068,4055,3960,4029,3839,4058,4070,4016,4059,3964,3584,4071,4074,4061,3966,4024,4075,4048,3967,4082,4028,4077,4083,4056,4030,3840,4085,4031,4060,4072,4089,4062,4076,4063,3968,4078,4084,4079,4086,4032,4087,4090,4091,4064,4093,4080,4088,4092,4094,4095,4096.

It can be easily learned that the corresponding first polarizationweight vector in which the polarization weights are sorted in ascendingorder and the corresponding first polarization weight vector in whichthe polarization weights are sorted in descending order are mutually ina reverse order. In other words, the former can be obtained bysequentially reading the latter in reverse order, or vice versa.

Both a first polarization weight vector that is corresponding to any Nor N_(max) and in which polarization weights are sorted in descendingorder and a first polarization weight vector that is corresponding toany N or N_(max) and in which polarization weights are sorted inascending order can be prestored.

FIG. 10 is a schematic structural diagram of an example of acommunications apparatus according to this application. Thecommunications apparatus may be a base station or a terminal. Based onthe same idea, this application provides a communications apparatus,configured to perform the foregoing method procedure. As shown in FIG.10, the apparatus 1000 includes a processor 1001, a memory 1002, and acommunications interface 1003. The processor 1001, the memory 1002, andthe communications interface 1003 are mutually coupled.

The memory 1002 may include a volatile memory (volatile memory), forexample, a RAM. The memory may alternatively include a non-volatilememory (non-volatile memory), for example, a flash memory (flashmemory), an HDD, or an SSD. The memory 1002 may alternatively include acombination of the foregoing types of memories.

The communications interface 1003 may be a wired communicationsinterface, a wireless communications interface, or a combinationthereof. The wired communications interface may be, for example, anEthernet interface. The Ethernet interface may be an optical interface,an electrical interface, or a combination thereof. The wirelesscommunications interface may be a WLAN interface.

The processor 1001 may be a CPU, an NP, or a combination of a CPU and anNP.

The processor 1001 may further include a hardware chip. The hardwarechip may be an ASIC, a PLD, or a combination thereof. The PLD may be aCPLD, an FPGA, a GAL, or any combination thereof. In some embodiments,the memory 1002 may be further combined with the processor 1001, forexample, may be implemented by using a circuit.

Optionally, the memory 1002 may be further configured to store aninstruction for performing various operations. The processor 1001invokes the instruction stored in the memory 1002, to perform one ormore steps or optional implementations in the foregoing embodiments, sothat the communications apparatus 1000 implements the foregoing methods,for example, calculates or stores at least one first polarization weightvector that is corresponding to N and in which polarization weights aresequentially sorted. A polarization weight of each polarized channel maybe calculated in a manner including but not limited to the methodsdisclosed in the foregoing embodiments.

All or some of the foregoing embodiments may be implemented by usingsoftware, hardware, firmware, or any combination thereof. When softwareis used to implement the embodiments, the embodiments may be implementedcompletely or partially in a form of a computer program product. Thecomputer program product includes one or more computer instructions.When the computer instructions are loaded and executed on the computer,the procedures or functions according to the embodiments of the presentinvention are all or partially generated. The computer may be ageneral-purpose computer, a dedicated computer, a computer network, oranother programmable apparatus. The computer instructions may be storedin a computer-readable storage medium or may be transmitted from acomputer-readable storage medium to another computer-readable storagemedium. For example, the computer instructions may be transmitted from awebsite, a computer, a server, or a data center to another website,computer, server, or data center in a wired (for example, a coaxialcable, an optical fiber, or a digital subscriber line (DSL)) or wireless(for example, infrared, radio, or microwave) manner. Thecomputer-readable storage medium may be any usable medium accessible bya computer, or a data storage device, such as a server or a data center,integrating one or more usable media. The usable medium may be amagnetic medium (for example, a floppy disk, a hard disk, or a magnetictape), an optical medium (for example, a DVD), a semiconductor medium(for example, a solid state disk (SSD)), or the like.

A person skilled in the art should understand that the embodiments ofthe present invention may be provided as a method or a computer programproduct. Therefore, the present invention may use a form of hardwareonly embodiments, software only embodiments, or embodiments with acombination of software and hardware. Moreover, the present inventionmay use a form of a computer program product that is implemented on oneor more computer-usable storage media (including but not limited to adisk memory, a CD-ROM, an optical memory, and the like) that includecomputer-usable program code.

The present invention is described with reference to the flowchartsand/or the block diagrams of the method, the device (system), and thecomputer program product according to the embodiments of the presentinvention. It should be understood that computer program instructionsmay be used to implement each process and/or each block in theflowcharts and/or the block diagrams, and a combination of a processand/or a block in the flowcharts and/or the block diagrams. Thesecomputer program instructions may be provided for a general-purposecomputer, a dedicated computer, an embedded processor, or a processor ofany other programmable data processing device to generate a machine, sothat the instructions executed by the computer or the processor of anyother programmable data processing device generate an apparatus forimplementing a specified function in one or more processes in theflowcharts and/or in one or more blocks in the block diagrams.

These computer program instructions may be stored in a computer readablememory that can instruct the computer or any other programmable dataprocessing device to work in a specific manner, so that the instructionsstored in the computer readable memory generate an artifact thatincludes an instruction apparatus. The instruction apparatus implementsa specified function in one or more processes in the flowcharts and/orin one or more blocks in the block diagrams.

These computer program instructions may alternatively be loaded onto acomputer or another programmable data processing device, so that aseries of operations and steps are performed on the computer or theanother programmable device, thereby generating computer-implementedprocessing. Therefore, the instructions executed on the computer or theanother programmable device provide steps for implementing a specifiedfunction in one or more processes in the flowcharts and/or in one ormore blocks in the block diagrams.

Although some embodiments of the present invention have been described,a person skilled in the art can make changes and modifications to theseembodiments once they learn of the basic inventive concept. Therefore,the following claims are intended to be construed as to cover theembodiments and all changes and modifications falling within the scopeof the present invention.

Obviously, a person skilled in the art can make various modificationsand variations to the present invention without departing from the scopeof the present invention. The present invention is intended to coverthese modifications and variations provided that they fall within thescope of protection defined by the following claims and their equivalenttechnologies.

What is claimed is:
 1. An encoding method, comprising: determining Nto-be-encoded bits, wherein the N to-be-encoded bits compriseinformation bits and frozen bits, and N is a positive integer; obtaininga first polarization weight vector comprising polarization weights of Npolarized channels, wherein the polarization weights are a basis forinformation bit selection, and the N to-be-encoded bits correspond tothe N polarized channels; determining positions of the information bitsbased on the first polarization weight vector; and performing polarencoding on the N to-be-encoded bits to obtain polar-encoded bits. 2.The method according to claim 1, wherein the obtaining a firstpolarization weight vector comprises: calculating the polarizationweights W_(i) of the N polarized channels according to the followingformula to obtain the first polarization weight vector:${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$wherein i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i, B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and n is apositive integer.
 3. The method according to claim 2, wherein ϕ=0, andα=¼.
 4. The method according to claim 1, wherein the obtaining a firstpolarization weight vector comprises: obtaining the first polarizationweight vector W₀ ^(N−1) through recursion calculation according to thefollowing formulas:β_(i)=(2^(i−1)+ϕ)^(α);W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹+β_(i); andW ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; whereW ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], whereiϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.
 5. The method according to claim 1, wherein a targetcode length M of a polar code is equal to a positive integer power of 2,and the obtaining a first polarization weight vector comprisingpolarization weights of N polarized channels comprises: obtaining thefirst polarization weight vector from a prestored second polarizationweight vector, wherein the second polarization weight vector comprisespolarization weights of N_(max) polarized channels, N_(max) is a presetmother code length corresponding to a maximum code length supported by acommunications system, and the mother code length is a positive integerpower of
 2. 6. The method according to claim 5, wherein: when the targetcode length M of the polar code is less than or equal to N_(max), anelement set in the first polarization weight vector is a subset of anelement set in the second polarization weight vector; or when the targetcode length M of the polar code is greater than or equal to N_(max), apolarization weight of a polarized channel whose channel index number isless than or equal to N_(max) is in a one-to-one correspondence with apolarization weight in the second polarization weight vector, and apolarization weight of a polarized channel whose channel index number isgreater than N_(max) is equal to a weight increment$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), wherein ϕand α are parameters that are preset based on the target code length anda code rate of the polar code.
 7. The method according to claim 1,wherein when the polarized channels in the first polarization weightvector are sorted in descending order of polarization weights, and aminimum sequence number value of the polarized channels is 1, if N=16,the first polarization weight vector is the following polarizationweight vector: 16,15,14,12,8,13,11,10,7,6,4,9,5,3,2,1; or if N=32, thefirst polarization weight vector is the following polarization weightvector:32,31,30,28,24,16,29,27,26,23,22,15,20,14,12,25,8,21,19,13,18,11,10,7,6,4,17,9,5,3,2,1;or if N=64, the first polarization weight vector is the followingpolarization weight vector:64,63,62,60,56,48,61,32,59,58,55,54,47,52,46,31,44,30,57,40,28,53,24,51,45,50,16,43,29,42,39,27,38,26,23,36,22,49,15,20,14,41,12,37,25,8,35,21,34,19,13,18,11,10,7,6,33,4,17,9,5, 3,2,1; or if N=128, the firstpolarization weight vector is the following polarization weight vector:128,127,126,124,120,112,125,96,123,122,119,64,118,111,116,110,95,108,94,104,92,63,117,62,88,115,109,60,114,80,107,56,93,106,103,91,48,102,90,61,87,100,32,59,86,113,79,58,84,55,78,105,54,76,47,101,52,89,46,72,99,31,44,85,98,30,57,83,40,77,28,82,53,75,24,51,74,45,71,50,16,43,70,97,29,42,68,39,27,81,38,26,23,36,73,22,49,15,69,20,14,41,67,12,66,37,25,8,35,21,34,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=256, thefirst polarization weight vector is the following polarization weightvector:256,255,254,252,248,240,253,224,251,250,247,192,246,239,244,238,128,223,236,222,249,232,220,191,245,190,216,243,237,188,242,127,208,235,184,126,221,234,231,124,219,176,230,218,189,120,215,228,160,187,214,241,112,207,186,212,183,125,206,233,182,96,123,204,175,229,180,122,217,174,119,200,227,64,159,118,172,213,226,111,158,185,116,211,168,110,205,156,210,181,95,108,203,152,179,94,121,202,173,104,199,178,92,63,144,117,171,198,225,62,157,88,170,115,196,167,109,60,155,114,209,166,80,107,15,884,156,751,164,937,106,201,103,150,177,91,48,143,102,197,148,90,61,142,87,169,100,195,32,59,86,140,113,194,165,79,58,153,84,55,136,163,78,105,54,149,162,76,47,101,52,147,89,46,141,72,99,146,31,44,85,139,98,193,30,57,138,83,40,135,77,28,82,53,134,161,75,24,51,132,74,45,71,50,145,16,43,70,97,29,42,137,68,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,2014,41,67,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=512, the firstpolarization weight vector is the following polarization weight vector:512,511,510,508,504,496,509,480,507,506,503,448,502,495,500,494,384,479,492,478,505,488,476,447,501,256,446,472,499,493,444,498,383,464,491,440,382,477,490,487,380,475,432,486,474,255,445,376,471,484,254,416,443,470,497,368,463,252,442,468,439,381,462,489,248,438,352,379,460,431,485,436,378,473,240,430,375,456,483,253,320,415,374,428,469,482,367,224,414,251,441,372,467,424,366,461,250,412,466,247,437,351,364,459,192,246,408,435,350,377,458,239,429,360,455,244,434,348,238,319,400,373,427,454,481,128,223,318,413,236,344,426,371,452,423,365,222,249,316,411,370,465,232,422,336,363,220,410,191,245,312,407,420,349,362,457,190,359,216,406,243,433,347,237,304,399,358,188,453,242,404,346,127,208,317,398,235,343,425,356,451,184,126,221,288,234,315,342,396,369,450,231,421,335,124,219,314,409,176,340,230,311,392,419,334361,218,189,120,215,310,405,228,418,332,303,160,357,187,214,241,308,403,345,112,207,302,397,328,186,355,212,402,183,125,206,287,233,300,341,395,354,449,182,96,286,123,204,313,394,175,339,229,296,391,180,333,122,217,284,174,338,119,200,309,390,227,417,331,64,159,118,172,213,280,226,307,388,330,111,301,158,327,1856,2,306,4068,160,205272,299,326,156,353,210,168,195,285,108,203,298,393,324,295,152,179,94,121,202,283,173,337,104,199,294,389,178,92,282,63,144,117,171,198,279,225,292,387,329,62,157,88,170,278,115,196,305,386,167,109,271,60,325,155,114,209,276,166,80,270,107,297,154,323,56,151,164,93,106,201,268,322,103,293,150,177,91,281,48,143,102,197,264,291,148,90,61,142,87,169,277,100,195,290,385,32,59,86,140,113,194,275,165,79,269,58,153,84,274,55,136,163,78,105,267,321,54,149,162,76,266,47,101,263,52,147,89,46,141,72,262,99,289,146,31,44,85,139,98,193,260,30,57,138,83,273,40,135,77,28,82,53,134,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=1024, the firstpolarization weight vector is the following polarization weight vector:1024,1023,1022,1020,1016,1008,1021,992,1019,1018,1015,960,1014,1007,1012,1006,896,991,1004,990,1017,1000,988,959,1013,768,958,984,1011,1005,956,1010,895,976,1003,952,894,989,1002,999,512,892,987,944,998,986,767,957,888,983,996,766,928,955,982,1009,880,975,764,954,980,951,893,974,1001,760,950,864,511,891,972,943,997,948,510,890,985,752,942,887,968,995,508,765,832,927,886,940,981,994,879,736,926,763,953,504,884,979,936,878,973,762,924,978,759,949,863,496,876,971,704,758,920,947,862,509,889,970,751,941,872,967,756,946,480,860,507,750,831,912,885,939,966,993,506,640,735,830,925,748,856,938,503,883,964,935,877,734,448,761,828,923,502,882,977,744,934,848,495,875,732,922,703,500,757,824,919,932,861,494,874,969,702,871,728,918,755,945,384,479,859,492,749,816,911,870,700,965,754,916,478,858,505,639,720,829,910,747,855,937,488,868,963,696,476,638,733,800,447,746,827,854,908,501,881,962,743,933,847,256,446,636,731,826,921,472,688,852,499,742,823,904,931,846,493,873,730,444,701,498,632,727,822,917,740,930,383,464,844,491,815,672,869,699,726,440,753,820,915,382,477,857,490,624,719,814,909,840,698,487,867,724,914,695,380,475,637,718,799,432,745,812,853,907,486,866,961,694,474,608,255,798,445,635,716,825,376,906,471,687,851,484,741,808,903,692,845,254,416,634,729,796,443,470,686,850,497,631,712,821,902,739,929,368,463,843,252,442,576,6710,468,630,684,725,792,439,738,819,900,381,462,842,489,623,813,670,839,697,248,438,628,723,818,913,352,680,379,460,622,717,784,431,811,838,668,485,865,722,436,693,378,473,607,240,797,430,620,715,810,375,905,456,836,483,807,664,691,606,253,320,415,633,714,795,374,428,469,685,849,482,616,711,806,901,690,367,224,414,604,251,794,441,575,656,372,467,629,683,710,791,424,737,804,899,366,461,841,250,412,574,669,466,600,682,247,790,437,627,708,817,898,351,679,364,459,621,783,192,572,837,667,246,408,626,721,788,435,350,678,377,458,592,239,782,429,619,809,360,666,455,835,244,434,568,663,348,676,605,238,319,400,618,713,780,373,427,454,834,481,615,805,662,689,128,223,318,413,603,236,793,344,426,560,655,371,452,614,709,776,423,803,660,365,222,602,249,316,411,573,654,370,465,599,681,232,789,422,612,707,802,897,336,363,220,410,544,191,571,598,652,245,312,407,625,706,787,420,349,677,362,457,591,781,190,570,359,665,216,406,596,243,786,433,567,648,347,675,590,237,304,399,617,779,358,188,453,833,242,404,566,661,346,674,127,208,317,398,588,235,778,343,425,559,356,451,613,775,184,564,659,126,221,288,601,234,315,342,396,558,653,369,450,584,231,774,421,611,801,658,335,124,219,314,409,543,176,340,556,597,651,230,311,392,610,705,772,419,334,361,218,542,189,569,120,650,215,310,405,595,228,785,418,552,647,332,589,303,160,540,357,187,214,594,241,308,403,565,646,345,673,112,207,302,397,587,777,328,186,355,212,402,536,183,563,644,125,206,287,586,233,300,341,395,557,354,449,583,773,182,562,657,96,286,123,204,313,394,528,175,339,555,582,229,296,391,609,771,180,333,122,217,284,541,174,338,554,119,649,200,309,390,580,227,770,417,551,331,64,159,539,118,172,213,280,593,226,307,388,550,645,330,111,301,158,538,327,185,116,211,306,401,535,168,548,643,110,205,272,585,299,326,156,353,210,534,181,561,642,95,285,108,203,298,393,527,324,581,295,152,532,179,94,121,202,283,526,173,337,553,104,199,294,389,579,769,178,92,282,63,144,524,117,171,198,279,578,225,292,387,549,329,62,157,537,88,170,278,115,196,305,386,520,167,547,109,271,60,325,155,114,209,276,533,166,546,641,80,270,107,297,154,323,56,151,531,164,93,106,201,268,525,322,103,293,150,530,177,91,281,48,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,32,59,86,140,113,194,275,518,165,545,79,269,58,153,84,274,55,136,516,163,78,105,267,321,54,149,529,162,76,266,47,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=2048,the first polarization weight vector is the following polarizationweight vector:2048,2047,2046,2044,2040,2032,2045,2016,2043,2042,2039,1984,2038,2031,2036,2030,1920,2015,2028,2014,2041,2024,2012,1983,2037,1792,1982,2008,2035,2029,1980,2034,1919,2000,2027,1976,1918,2013,2026,2023,1536,1916,2011,1968,2022,2010,1791,1981,1912,2007,2020,1790,1952,1979,2006,2033,1904,1999,1788,1978,2004,1975,1917,1998,2025,1024,1784,1974,1888,1535,1915,1996,1967,2021,1972,1534,1914,2009,1776,1966,1911,1992,2019,1532,1789,1856,1951,1910,1964,2005,2018,1903,1760,1950,1787,1977,1528,1908,2003,1960,1902,1997,1786,1948,1023,2002,1783,1973,1887,1520,1900,1995,1728,1022,1782,1944,1971,1886,1533,1913,1994,1775,1965,18961991020,10,1970,1504,1884,1531,1774,1855,1936,1909,1963,1990,2017,1016,1530,1664,1759,1854,1949,1772,1880,1962,1527,1907,1988,1959,1901,1758,1472,1785,1852,1947,1526,1906,1008,2001,1768,1958,1872,1519,1899,1756,1946,1727,1021,1524,1781,1848,1943,1956,1885,1518,1898,1993,1726,992,1895,1752,1019,1942,1779,1969,1408,1503,1883,1516,1773,1840,1935,1894,1724,1989,1018,1778,1940,1502,1015,1882,1529,1663,1744,1853,1934,1771,1879,1961,1512,1892,1987,9601720,1014,1500,1662,1757,1824,1471,1770,1851,1878,1932,1525,1905,1007,1986,1767,1957,1871,1012,1280,1470,1660,1755,1850,1945,1496,1712,1876,1006,1523,1766,1847,1928,1955,1870,1517,1897,1754,1468,1725,896,991,1522,1656,1751,1846,1004,1941,1764,1954,1407,1488,1868,1515,1839,1696,990,1893,1723,1750,1017,1464,1777,1844,1939,1406,1501,1000,1881,1514,1648,1743,1838,1933,1864,1722,1511,988,1891,1748,959,1938,1719,1013,1404,1499,1661,1742,1823,1456,1769,1836,1877,1931,1510,1890,768,1985,958,1718,984,1498,1632,1011,1279,1822,1469,1659,1740,1849,1400,1930,1495,1711,1875,1005,1508,1765,1832,1927,956,1716,1869,1010,1278,1440,1658,1753,1820,1467,1494,1710,895,1874,976,1521,1655,1736,1845,1003,1926,1763,1953,1392,1487,1867,952,1276,1466,1600,1695,894,989,1492,1654,1708,1749,1816,1002,1463,1762,1843,1924,1405,1486,999,1866,1513,1647,1837,1694,512,1863,1721,892,1272,987,1462,1652,1747,1842,944,1937,1376,1704,998,1403,1484,1646,1741,1808,1455,1835,1862,1692,1509,986,1889,767,1746,957,1461,710,7888,983,1402,1497,1631,996,1264,1821,1454,1644,1739,1834,1399,1929,1480,1860,766,1507,928,1831,1688,955,982,1715,1630,1009,1277,1344,1439,1657,1738,1819,1398,1452,1493,1709,880,1873,975,1506, 1640,1735,1830,764,1925,954,1714,1391,980,1248,1438,1628,951,1275,1818,1465,1599,1680,893,1396,974,1491,1653,1707,1734,1815,1001,1448,1761,1828,1923,1390,1485,760,1865,950,1274,1436,1598,1693,864,511,1491624,90,624,1706,891,1271,1814,972,1461,1651,1732,1841,943,1922,1375,1703,997,1388,1483,1645,1807,948,1216,510, 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7,6,33,4,17,9,5,3,2,1; orif N=4096, the first polarization weight vector is the followingpolarization weight 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2190,697,248,1353,2570,2135,438,628,2217,2325,723,818,1086,2148,913,1181,352,2243,2338,680,1561,2433,1112,2567,1194,379,1302,3081,460,622,1139,717,784,1220,1329,431,1410,1544,2080,811,1191,838,668,485,1571,2107,2134,865,2188,1133,722,1295,2161,436,2242,2323,693,1084,1349,2566,1179,378,2213,473,607,1138,2593,240,797,1233,1300,2127,430,620,1557,2317,715,810,1190,375,905,3077,456,1570,2106,1665,836,2201,1104,483,1294,2132,807,1131,2322,664,1321,2103,1178,2184,691,606,1347,2564,253,320,2211,2126,415,633,1080,714,795,1175,374,2153,428,469,1555,2315,685,1188,849,1117,3075,482,616,2369,711,806,1130,901,1225,1292,2102,1549,2197,690,367,1346,2577,224,2210,1127,414,604,1317,251,794,1174,2124,441,575,1554,2314,656,1201,372,2095,3074,467,629,683,710,791,1115,2149,424,1305,2311,737,804,1072,899,1167,366,2100,461,1547,2195,1126,841,1221,1288,250,1601,2137,412,574,1315,669,1172,2094,466,600,2189,682,247,790,1114,2569,2120,437,627,2310,708,817,1085,2147,898,1166,351,2337,679,1546,2194,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,2079,572,837,667,246,2092,2133,408,626,2187,721,788,1056,2146,435,350,2241,2308,678,1083,1110,2565,1164,377,458,592,137,239,782,1218,1299,429,619,1542,2078,809,1189,360,666,455,1569,2105,835,2186,1103,244,1293,2131,434,568,2321,663,1082,2088,1177,348,2183,676,605,1108,2563,238,319,1298,2125,400,618,1079,713,780,1160,373,427,454,1540,2076,1187,834,1102,481,615,2130,805,1129,662,1291,2101,2182,689,128,1345,2562,223,318,2209,413,603,1078,236,793,1173,344,2123,426,560,1553,2313,655,1186,371,3073,452,614,709,776,1100,423,1290,2072,803,1071,660,365,2099,222,2180,1125,602,1287,249,316,2122,411,573,1076,654,1171,370,2093,465,599,681,232,789,1113,2119,422,612,2309,707,802,1070,897,1165,336,2098,1545,2193,1096,363,1286,220,1123,410,544,1313,191,1170,2064,571,598,652,245,312,2091,2118,407,625,706,787,1055,2145,420,349,2307,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,1541,2077,570,359,665,216,2090,406,596,2185,243,786,1054,2116,433,567,2306,648,1081,2087,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,2075,833,1101,242,2129,404,566,661,1052,2086,346,2181,674,127,1106,2561,208,317,398,588,1077,235,778,1158,343,425,559,1538,2074,1185,356,451,613,775,1099,184,1289,2071,564,659,126,2084,221,288,2179,601,1048,234,315,342,2121,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,2070,801,1069,658,335,2097,2178,1095,124,1285,219,314,409,543,1074,176,1169,340,2063,556,597,651,230,311,2117,392,610,705,772,1040,419,334,2068,1067,1094,361,218,1121,542,1283,189,2062,569,120,650,215,310,2089,405,595,228,785,1053,2115,418,552,2305,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,2060,594,241,308,2114,403,565,646,1051,2085,345,673,112,1105,207,302,397,587,1062,777,1157,328,186,1537,2073,355,212,402,536,183,1050,2056,563,644,125,2083,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,2069,562,657,96,2082,286,2177,123,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,180,333,2067,1093,122,217,284,541,1044,174,338,2061,554,119,649,200,309,390,580,227,770,1038,417,551,2066,1065,331,1091,64,1281,159,539,118,172,213,280,2059,593,226,307,2113,388,550,645,1036,330,111,1090,301,158,1061,538,327,185,2058,116,211,306,401,535,168,1049,2055,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,2054,561,642,95,2081,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,2052,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,2065,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,2057,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,2053,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,2051,106,201,268,525,1028,322,103,293,150,530,177,2050,91,281,48,1041,143,523,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,2049,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or when a minimum sequence number in the first polarizationweight vector is 0, 1 is subtracted from sequence numbers in the firstpolarization weight vector successively.
 8. The method according toclaim 1, wherein when a polarization weight of a first polarized channelis greater than that of a second polarized channel, the first polarizedchannel is before the second polarized channel in the first polarizationweight vector.
 9. A communications apparatus, comprising: a memory, thememory configured to store an instruction for performing an operation;and at least one processor, the at least one processor coupled with thememory, and according to the instruction, configured to: determine Nto-be-encoded bits, wherein the N to-be-encoded bits compriseinformation bits and frozen bits, and N is a positive integer; obtain afirst polarization weight vector comprising polarization weights of Npolarized channels, wherein the polarization weights are a basis forinformation bit selection, and the N to-be-encoded bits correspond tothe N polarized channels; determine positions of the information bitsbased on the first polarization weight vector; and perform polarencoding on the N to-be-encoded bits to obtain polar-encoded bits. 10.The communications apparatus according to claim 9, wherein the at leastone processor is configured to: calculate the polarization weights W_(i)of the N polarized channels according to the following formula to obtainthe first polarization weight vector:${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$wherein i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i, B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and n is apositive integer.
 11. The communications apparatus according to claim10, wherein ϕ=0, and α=¼.
 12. The communications apparatus according toclaim 9, wherein the at least one processor is configured to: obtain thefirst polarization weight vector W₀ ^(N−1) through recursion calculationaccording to the following formulas:β_(i)=(2^(i−1)+ϕ)^(α);W ₂ _(i−1) ² ^(i) ⁻¹ =W ₀ ² ^(i−1) ⁻¹+β_(i); andW ₀ ² ^(i) ⁻¹=[W ₀ ² ^(i−1) ⁻¹ ,W ₂ _(i−1) ² ^(i) ⁻¹]; whereW ₀ ¹=[W ₀ ,W ₁],W ₀ ² ^(i) ⁻¹==[W ₀ ,W ₁ , . . . ,W ₂ _(i−1) ], whereiϵ{1, . . . , N−1}, β_(i) is an intermediate variable related to achannel index number i, ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and W₀ is anarbitrary constant.
 13. The communications apparatus according to claim9, wherein a target code length M of a polar code is equal to a positiveinteger power of 2, and the at least one processor is configured to:obtain the first polarization weight vector from a prestored secondpolarization weight vector, wherein the second polarization weightvector comprises polarization weights of N_(max) polarized channels,N_(max) is a preset mother code length corresponding to a maximum codelength supported by a communications system, and the mother code lengthis a positive integer power of
 2. 14. The communications apparatusaccording to claim 13, wherein: when the target code length M of thepolar code is less than or equal to N_(max), an element set in the firstpolarization weight vector is a subset of an element set in the secondpolarization weight vector; or when the target code length M of thepolar code is greater than or equal to N_(max), a polarization weight ofa polarized channel whose channel index number is less than or equal toN_(max) is in a one-to-one correspondence with a polarization weight inthe second polarization weight vector, and a polarization weight of apolarized channel whose channel index number is greater than N_(max) isequal to a weight increment$\beta = {\sum\limits_{j = {\log_{2}{(N_{\max})}}}^{\lfloor{\log_{2}{({M - 1})}}\rfloor}( {2^{j} + \varphi} )^{\alpha}}$plus the corresponding polarization weight of the polarized channelwhose channel index number is less than or equal to N_(max), wherein ϕand α are parameters that are preset based on the target code length anda code rate of the polar code.
 15. The communications apparatusaccording to claim 9, wherein when the polarized channels in the firstpolarization weight vector are sorted in descending order ofpolarization weights, and a minimum sequence number value of thepolarized channels is 1, if N=16, the first polarization weight vectoris the following polarization weight vector:16,15,14,12,8,13,11,10,7,6,4,9,5,3,2,1; or if N=32, the firstpolarization weight vector is the following polarization weight vector:32,31,30,28,24,16,29,27,26,23,22,15,20,14,12,25,8,21,19,13,18,11,10,7,6,4,17,9,5,3,2,1;or if N=64, the first polarization weight vector is the followingpolarization weight vector:64,63,62,60,56,48,61,32,59,58,55,54,47,52,46,31,44,30,57,40,28,53,24,51,45,50,16,43,29,42,39,27,38,26,23,36,22,49,15,20,14,41,12,37,25,8,35,21,34,19,13,18,11,10,7,6,33,4,17,9,5,3,2,1; or if N=128, the first polarization weight vector is thefollowing polarization weight vector:128,127,126,124,120,112,125,96,123,122,119,64,118,111,116,110,95,108,94,121,104,92,63,117,62,88,115,109,60,4,80,10607,56,93,14,8010756,93106,103,91,48,102,90,61,87,100,3259,86,113,79,58,84,55,78,105,54,76,47,101,52,89,46,72,99,31,44,85,98,30,57,83,40,77,28,82,53,75,24,51,74,45,71,50,16,43,70,97,29,42,68,39,27,81,38,26,23,36,73,22,49,15,69,20,14,41,67,12,66,37,25,8,35,21,34,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=256, thefirst polarization weight vector is the following polarization weightvector:256,255,254,252,248,240,253,224,251,250,247,192,246,239,244,238,128,223,236,222,249,232,220,191,245,190,216,243,237,188,242,127,208,235,184,126,221,234,231,124,219,176,230,218,189,120,215,228,160,187,214,241,112,207,186,212,183,125,206,233,182,96,123,204,175,229,180,122,217,174,119,200,227,64,159,118,172,213,226,111,158,185,116,211,168,110,205,156,210,181,95,108,203,152,179,94,121,202,173,104,199,178,92,63,144,117,171,198,225,62,157,88,170,115,196,167,109,60,155,114,209,166,80,107,154,56,151,164,93,106,201,103,150,177,91,48,143,102,197,148,90,61,142,87,169,100,195,32,59,86,140,113,194,165,79,58,153,84,55,136,163,78,105,54,149,162,76,47,101,52,147,89,46,141,72,99,146,31,44,85,139,98,193,30,57,138,83,40,135,77,28,82,53,134,161,75,24,51,132,74,45,71,50,145,16,43,70,97,29,42,137,68,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=512, the firstpolarization weight vector is the following polarization weight vector:512,511,510,508,504,496,509,480,507,506,503,448,502,495,500,494,384,479,492,478,505,488,476,447,501,256,446,472,499,493,444,498,383,464,491,440,382,477,490,487,380,475,432,486,474,255,445,376,471,4842541,416,443,470,497,368,463252,442,468439,381,462,489,248,438,352,379,460,431,485,436,378,473,240,430,375,456,483,253,320,415,374,428,469,482,367,224,414,251,441,372,467,424,366,461,250,412,466,247,437,351,364,459,192,246,408,435,350,377,458,239,429,360,455,244,434,348,238,319,400,373,427,454,481,128,223,318,413,236,344,426,371,452,423,365,222,249,316,411,370,465,232,422,336,363,220,410,191,245,312,407,420,349,362,457,190,359,216,406,243,433,347,237,304,399,358,188,453,242,404,346,127,208,317,398,235,343,425,356,451,184,126,221,288,234,315,342,396,369,450,231,421,335,124,219,314,409,176,340,230,311,392,419,334,361,218,189,120,215,310,405,228,418,332,303,160,357,187,214,241,308,403,345,112,207,302,397,328,186,355,212,402,183,125,206,287,233,300,341,395,354,449,182,96,286,123,204,313,394,175,339,229,296,391,180,333,122,217,284,174,338,119,200,309,390,227,417,331,64,159,118,172,213,280,226,307,388,330,111,301,158,327,185,116,211,306,401,168,110,205,272,299,326,156,353,210,181,95,285,108,203,298,393,324,295,152,179,94,121,202,283,173,337,104,199,294,389,178,92,282,63,144,117,171,198,279,225,292,387,329,62,157,88,170,278,115,196,305,386,167,109,271,60,325,155,114,209,276,166,80,270,107,297,154,323,56,151,164,93,106,201,268,322,103,293,150,177,91,281,48,143,102,197,264,291,148,90,61,142,87,169277,100,195,290,385,32,59,86,140,113,194,275,165,79,269,58,153,84,274,55,136,163,78,105,267,321,54,149,162,76,266,47,101,263,52,147,89,46,141,72,262,99,289,146,31,44,85,139,98,193,260,30,57,138,83,273,40,135,77,28,82,53,134,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=024, the firstpolarization weight vector is the following polarization weight vector:1024,1023,1022,1020,1016,1008,1021,992,1019,1018,1015,960,1014,1007,1012,1006,896,991,1004,990,1017,1000,988,959,1013,768,958,984,1011,1005,956,1010,895,976,1003,952,894,989,1002,999,512,892,987,944,998,986,767,957,888,983,996,766,928,955,982,1009,880,975,764,954,980,951,893,974,1001,760,950,864,511,891,972,943,997,948,510,890,985,752,942,887,968,995,508,765,832,927,886,940,981,994,879,736,926,763,953,504,884,979,936,878,973,762,924,978,759,949,863,496,876,971,704,758,920,947,862,509,889,970,751,941,872,967,756,946,480,860,507,750,831,912,885,939,966,993,506,640,735,830,925,748,856,938,503,883,964,935,877,734,448,761,828,923,502,882,977,744,934,848,495,875,732,922,703,500,757,824,919,932,861,494,874,969,702,871,728,918,755,945,384,479,859,492,749,816,911,870,700,965,754,916,478,858,505,639,720,829,910,747,855,937,488,868,963,696,476,638,733,800,447,746,827,854,908,501,881,962,743,933,847,256,446,636,731,826,921,472,688,852,499,742,823,904,931,846,493,873,730,444,701,498,632,727,822,917,740,930,383,464,844,491,815,672,869,699,726,440,753,820,915,382,477,857,490,624,719,814,909,840,698,487,867,7 24,914,695,380,475,637,718,799,432,745,812,853,907,486,866,961,694,474,608,255,798,445,635,716,825,376,906,471,687,851,484,741,808,903,692,845,254,416,634,729,796,443,470,686,850,497,631,712,821,902,739,929,368,463,843,252,442,576,671,468,630,684,725,792,439,738,819,900,381,462,842,489,623,813,670,839,697,248,438,628,723,818,913,352,680,379,460,622,717,784,431,811,838,668,485,865,722,436,693,378,473,607,240,797,430,620,715,810,375,905,456,836,483,807,664,691,606,253,320,415,633,714,795,374,428,469,685,849,482,616,711,806,901,690,367,224,414,604,251,794,441,575,656,372,467,629,683,710,791,424,737,804,899,366,461,841,250,412,574,669,466,600,682,247,790,437,627708,817,898,351,679,364,459,621,783,192,572,837,667,246,408,626,721,788,435,350,678,377,458,592,239,782,429,619,809,360,666,455,835,244,434,568,663,348,676,605,238,319,400,618,713,780,373,427,454,834,481,615,805,662,689,128,223,318,413,603,236,793,344,426,560,655,371,452,614,709,776,423,803,660,365,222,602,249,316,411,573,654,370,465,599,681,232,789,422,612,707,802,897,336,363,220,410,544,191,571,598,652,245,312,407,625,706,787,420,349,677,362,457,591,781,190,570,359,665,216,406,596,243,786,433,567,648,347,675,590,237,304,399,617,779,358,188,453,833,242,404,566,661,346,674,127,208,317,398,588,235,778,343,425,559,356,451,613,775,184,564,659,126,221,288,601,234,315,342,396,558,653,369,450,584,231,774,421,611,801,658,335,124,219,314,409,543,176,340,556,597,651,230,311,392,610,705,772,419,334,361,218,542,189,569,120,650,215,310,405,595,228,785,418,552,647,332,589,303,160,540,357,187,214,594,241,308,403,565,646,345,673,112,207,302,397,587,777,328,186,355,212,402,536,183,563,644,125,206,287,586,233,300,341,395,557,354,449,583,773,182,562,657,96,286,123,204,313,394,528,175,339,555,582,229,296,391,609,771,180,333,122,217,284,541,174,338,554,119,649,200,309,390,580,227,770,417,551,331,64,159,539,118,172,213,280,593,226,307,388,550,645,330,111,301,158,538,327,185,116,211,306,401,535,168,548,643,110,205,272,585,299,326,156,353,210,534,181,561,642,95,285,108,203,298,393,527,324,581,295,152,532,179,94,121,202,283,526,173,337,553,104,199,294,389,579,769,178,92,282,63,144,524,117,171,198,279,578,225,292,387,549,329,62,157,537,88,170,278,115,196,305,386,520,167,547,109,271,60,325,155,114,209,276,533,166,546,641,80,270,107,297,154,323,56,151,531,164,93,106,201,268,525,322,103,293,150,530,177,91,281,48,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,32,59,86,140,113,194,275,518,165,545,79,269,58,153,84,274,55,136,516,163,78,105,267,321,54,149,529,162,76,266,47,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,5175,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=2048,the first polarization weight vector is the following polarizationweight vector:2048,2047,2046,2044,2040,2032,2045,2016,2043,2042,2039,1984,2038,2031,2036,2030,1920,2015,2028,2014,2041,2024,2012,1983,2037,1792,1982,2008,2035,2029,1980,2034,1919,2000,2027,1976,1918,2013,2026,2023,1536,1916,2011,1968,2022,2010,1791,1981,1912,2007,2020,1790,1952,1979,2006,2033,1904,1999,1788,1978,2004,1975,1917,1998,2025,1024,1784,1974,1888,1535,1915,1996,1967,2021,1972,1534,1914,2009,1776,1966,191,1992,2019,1532,1789,1856,1951,1910,1964,2005,2018,1903,1760,1950,1787,1977,1528,1908,2003,1960,1902,1997,1786,1948,1023,2002,1783,1973,1887,1520,1900,1995,1728,1022,1782,1944,1971,1886,1533,1913,1994,1775,1965,1896,1991,1020,1780,1970,1504,1884,1531,1774,1855,1936,1909,1963,1990,2017,1016,1530,1664,1759,1854,1949,1772,1880,1962,1527,1907,1988,1959,1901,1758,1472,1785,1852,1947,1526,1906,1008,2001,1768,1958,1872,1519,1899,1756,1946,1727,1021,1524,1781,1848,1943,1956,1885,1518,1898,1993,1726,992,1895,1752,1019,1942,1779,1969,1408,1503,1883,1516,1773,1840,1935,1894,1724,1989,1018,1778,1940,1502,1015,1882,1529,1663,1744,1853,1934,1771,1879,1961,1512,1892,1987,960,1720,1014,1500,1662,1757,1824,1471,1770,1851,1878,1932,1525,1905,1007,1986,1767,1957,1871,1012,1280,1470,1660,1755,1850,1945,1496,1712,1876,1006,1523,1766,1847,1928,1955,1870,1517,1897,1754,1468,1725,896,991,1522,1656,1751,1846,1004,1941,1764,1954,1407,1488,1868,1515,1839,1696,990,1893,1723,1750,1017,1464,1777,1844,19391,11777,1844,1939,1406,1501,1000,1881,1514,1648,1743,1838,1933,1864,1722,1511,988,1891,1748,959,1938,1719,1013,1404,1499,1661,1742,1823,1456,1769,1836,1877,193,1510,1890,7681985,958,1718,9841,4981,632,101,151,1279,1822,1469,1659,1740,1849,1400,1930,1495,1711,1875,1005,1508,1765,1832,1927,956,1716,1869,1010,1278,1440,1658,1753,1820,1467,1494,1710,895,1874,976,1521,1655,1736,1845,1003,1926,1763,1953,1392,1487,1867,952,1276,1466,1600,1695,894,989,1492,1654,1708,1749,1816,1002,1,1463,1762,1843,1924,1405,1486,999,1866,1513,1647,1837,1694,512,1863,1721,892,1272,987,1462,1652,1747,1842,944,1937,1376,1704,998,1403,1484,1646,1741,1808,1455,1835,1862,1692,1509,986,1889,767,1746,957,1460,1717,888,983,1402,1497,1631,996,1264,1821,1454,1644,1739,1834,1399,1929,1480,1860,766,1507,928,1831,1688,955,982,1715,1630,1009,1277,1344,1439,1657,1738,1819,1398,1452,1493,1709,880,1873,975,1506,1640,1735,1830,764,1925,954,1714,1391,980,1248,1438,1628,951,1275,1818,1465,1599,1680,893,1396,974,1491,1653,1707,1734,1815,1001,1448,1761,1828,1923,1390,1485,760,1865,950,1274,1436,1598,1693,864,511,1490,1624,1706,891,1271,1814,972,1461,1651,1732,1841,943,1922,1375,1703,997,1388,1483,1645,1807,948,1216,510,1596,1861,1691,890,1270,985,1432,1650,752,1745,1812,942,1459,1374,1702,887,968,1401,1482,1616,995,1263,1806,1453,1643,1833,1384,1690,1479,508,1859,765,832,1268,927,1458,1592,1687,886,940,981,1372,1700,1629,994,1262,1343,1424,1642,1737,1804,1397,1451,1478,879,1858,736,1505,1639,926,1829,763,1686,953,504,1713,884,1152,979,1247,1342,1437,1627,936,1260,1817,1368,1450,1584,1679,878,1395,973,1476,1638,1733,1800,762,1447,924,1827,1684,1389,978,1246,759,1626,949,1273,1340,1435,1597,1678,863,1394,496,1489,1623,1705,876,1256,1813,971,1446,1636,1731,1826,704,1921,1360,758,138792124244,1434,1568,947,1215,862,509,1595,1622,1676,889,1269,1336,970,1431,1649,751,1730,1811,941,1444,1373,1701,872,967,1386,1481,1615,756,1805,946,1214,480,1594,1383,1689,860,1240,507,1430,1620,750,831,1267,1810,912,1457,1591,1672,885,939,966,1371,1699,1614,993,1261,1328,1423,1641,1803,1382,1212,1477,506,640,1857,735,830,1266,925,1428,1590,748,1685,856,938,503,1370,1698,883,1151,964,1232,1341,1422,1612,935,1259,1802,1367,1449,1583,877,1380,734,1475,1637,448,1799,761,828,1208,923,502,1588,1683,882,1150,977,1245,1312,744,1625,934,1258,1339,1366,1420,1582,1677,848,1393,495,1474,1608,875,1255,1798,732,1445,1635,922,1825,703,1682,1359,500,757,824,1148,919,1243,1338,1433,1567,932,1200,861,1364,494,1580,1621,1675,874,1254,1335,969,1416,1634,1729,1796,702,1443,1358,871728,1385,918,1242,755,1566,945,1213,384,479,1593,1144,1674,859,1239,1334,492,1429,1619,749,816,1252,1809,911,1442,1576,1671,870,700,965,1356,1613,754,1327,916,1184,478,1564,1381,1211,858,1238,505,639,1618,720,829,1265,1332,910,1427,1589,747,1670,855,937,488,1369,1697,868,1136,963,1231,1326,1421,1611,696,1801,1352,1210,476,638,1379,733,800,1236,447,1426,1560,746,827,1207,854,908,501,1587,1668,881,1149,962,1230,1311,743,1610,933,1257,1324,1365,1419,1581,847,1378,256,147311607,44646,636,1797,731,826,1206,921,472,1586,688,1681,852,1120,499,1310,742,823,1147,904,1228,1337,1418,1552,931,199,846,1363,493,1579,1606,873,1253,1320,730,1415,1633,444,1795,701,1204,1357,498,632,727,822,1146,917,1241,1308,740,1565,930,1198,383,1362,464,1578,1143,1673,844,1224,1333,491,1414,1604,815,1251,1794,672,1441,1575,869,699,726,1355,440,753,820,1088,915,1183,382,477,1563,1142,1196,857,1237,1304,490,624,1617,719,814,1250,1331,909,1412,1574,1669,840,698,487,1354,867,1135,724,1325,914,1182,695,1562,1351,1209,380,475,637,1140,718,799,1235,1330,432,1425,1559,745,812,1192,853,907,486,1572,1667,866,1134,961,1229,1296,1609,694,1323,1350,1180,474,608,1377,255,798,1234,445,635,1558,716,825,1205,376,906,471,1585,687,1666,851,1119,484,1309,741,808,1132,903,1227,1322,1417,1551,692,845,1348,254,1605,416,634,1319,729,796,1176,443,470,1556,686,1203,850,1118,497,631,712,821,1145,902,1226,1307,739,1550,929,1197,368,1361,463,1577,1128,843,1223,1318,252,1413,1603,442,576,1793,671,1202,468,630,684,725,792,1116,439,1306,738,819,1087,900,1168,381,462,1548,1141,1195,842,1222,1303,489,623,1602,813,1249,1316,670,1411,1573,839,697,248,1353,438,628,723,818,1086,913,1181,352,680,1561,1112,1194,379,1302,460,622,1139,717,784,1220,1329,431,1410,1544,811,1191,838,668,485,1571,865,1133,722,1295,436,693,1084,1349,1179,378,473,607,1138,240,797,1233,1300,430,620,1557,715,810,1190,375,905,456,1570,1665,836,1104,483,1294,807,1131,664,1321,1178,691,606,1347,253,320,415,633,1080,714,795,1175,374,428,469,1555,685,1188,849,1117,482,616,711,806,1130,901,1225,1292,1549,690,367,1346,224,1127,414,604,1317,251,794,1174,441,575,1554,656,1201,372,467,629,683,710,791,1115,424,1305,737,804,1072,899,1167,366,461,1547,1126,841,1221,1288,250,1601,412,574,1315,669,1172,466,600,682,247,790,1114,437,627,708,817,1085,898,1166,351,679,1546,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,572,837,667,246,408,626,721,788,1056,435,350,678,1083,1110,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,809,1189,360,666,455,1569,835,1103,244,1293,434,568,663,1082,1177,348,676,605,1108,238,319,1298,400,618,1079,713,780,1160,373,427,454,1540,1187,834,1102,481,615,805,1129,662,1291,689,128,1345,223,318,413,603,1078,236,793,1173,344,426,560,1553,655,1186,371,452,614,709,776,1100,423,1290,803,1071,660,365,222,1125,602,1287,249,316,411,573,1076,654,1171,370,465,599,681,232,789,1113,422,612,707,802,1070,897,1165,336,1545,1096,363,1286,220,1123,410,544,1313,191,1170,571,598,652,245,312,407,625,706,787,1055,420,349,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,1541,570,359,665,216,406,596,243,786,1054,433,567,648,1081,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,833,1101,242,404,566,661,1052,346,674,127,1106,208,317,398,588,1077,235,778,1158,343,425,559,1538,1185,356,451,613,775,1099,184,1289,564,659,126,221,288,601,1048,234,315,342,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,801,1069,658,335,1095,124,1285,219,314,409,543,1074,176,1169,340,556,597,651,230,311,392,610,705,772,1040,419,334,1067,1094,361,218,1121,542,1283,189,569,120,650,215,310,405,595,228,785,1053,418,552,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,594,241,308,403,565,646,1051,345,673,112,1105,207,302,397,587,1062,777,1157,328,186,1537,355,212,402,536,183,1050,563,644,125,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,562,657,96,286,123,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,3391,609,771,1039,180,333,1093,122,217,284,541,1044,174,338,554,119,649,200,309,390,580,227,770,1038,417,551,1065,331,1091,64,1281,159,539,118,172,213,280,593,226,307,388,550,645,1036,330,111,1090,301,158,1061,538,327,185,116,211,306,401,535,168,1049,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,561,642,95,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,106,201,268,525,1028,322,103,293,150,530,177,91,281,48,1041,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; orif N=4096, the first polarization weight vector is the followingpolarization weight 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109,839,2190,697,248,1353,2570,2135,438,628,2217,2325,723,818,1086,2148,913,1181,352,2243,2338,680,1561,2433,1112,2567,1194,379,1302,3081,460,622,1139,717,784,1220,1329,431,1410,1544,2080,811,1191,838,668,485,1571,2107,2134,865,2188,1133,722,1295,2161,436,2242,2323,693,1084,1349,2566,1179,378,2213,473,607,1138,2593,240,797,1233,1300,2127,430,620,1557,2317,715,810,1190,375,905,3077,456,1570,2106,1665,836,2201,1104,483,1294,2132,807,1131,2322,664,1321,2103,1178,2184,691,606,1347,2564,253,320,2211,2126,415,633,1080,714,795,1175,374,2153,428,469,1555,2315,685,1188,849,1117,3075,482,616,2369,711,806,1130,901,1225,1292,2102,1549,2197,690,367,1346,2577,224,2210,1127,414,604,1317,251,794,1174,2124,441,575,1554,2314,656,1201,372,2095,3074,467,629,683,710,791,1115,2149,424,1305,2311,737,804,1072,899,1167,366,2100,461,1547,2195,1126,841,1221,1288,250,1601,2137,412,574,1315,669,1172,2094,466,600,2189,682,247,790,1114,2569,2120,437,627,2310,708,817,1085,2147,898,1166,351,2337,679,1546,2194,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,2079,572,837,667,246,2092,2133,408,626,2187,721,788,1056,2146,435,350,2241,2308,678,1083,1110,2565,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,2078,809,1189,360,666,455,1569,2105,835,2186,1103,244,1293,2131,434,568,2321,663,1082,2088,1177,348,2183,676,605,1108,2563,238,319,1298,2125,400,618,1079,713,780,1160,373,427,454,1540,2076,1187,834,1102,481,615,2130,805,1129,662,1291,2101,2182,689,128,1345,2562,223,318,2209,413,603,1078,236793,1173,344,2123,426,560,1553,2313,655,1186,371,3073,452,614,709,776,1100,423,1290,2072,803,1071,660,365,2099,222,2180,1125,602,1287,249,316,2122,411,573,1076,654,1171,370,2093,465,599,681,232,789,1113,2119,422,612,2309,707,802,1070,897,1165,336,2098,1545,2193,1096,363,1286,220,1123,410,544,1313,191,1170,2064,571,598,652,245,312,2091,2118,407,625,706,787,1055,2145,420,349,2307,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,1541,2077,570,359,665,216,2090,406,596,2185,243,786,1054,2116,433,567,2306,648,1081,2087,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,2075,833,1101,242,2129,404,566,661,1052,2086,346,2181,674,127,1106,2561,208,317,398,588,1077,235,778,1158,343,425,559,1538,2074,1185,356,451,613,775,1099,184,1289,2071,564,659,126,2084,221,288,2179,601,1048,234,315,342,2121,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,2070,801,1069,658,335,2097,2178,1095,124,1285,219,314,409,543,1074,176,1169,340,2063,556,597,651,230,311,2117,392,610,705,772,1040,419,334,2068,1067,1094,361,218,1121,542,1283,189,2062,569,120,650,215,310,2089,405,595,228,785,1053,2115,418,552,2305,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,2060,594,241,308,2114,403,565,646,1051,2085,345,673,112,1105,207,302,397,587,1062,777,1157,328186,1537,2073,355,212,402,536,183,1050,2056,563,644,125,2083,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,2069,562,657,96,2082,286,2177,123,1046204313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,180,333,2067,1093,122,217,284,541,1044,174,338,2061,554,119,649,200,309,390,580,227,770,1038,417,551,2066,1065,331,1091,64,1281,159,539,118,172,213,280,2059,593,226,307,2113,388,550,645,1036,330,111,1090,301,158,1061,538,327,185,2058,116,211,306,401,535,168,1049,2055,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,2054,561,642,95,2081,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,2052,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,2065,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,2057,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,2053,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,2051,106,201,268,525,1028,322,103,2930,53177,772050,91281,48,1041,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,2049,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,14631,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or when a minimum sequence number in the first polarizationweight vector is 0, 1 is subtracted from sequence numbers in the firstpolarization weight vector successively.
 16. The communicationsapparatus according to claim 9, wherein when a polarization weight of afirst polarized channel is greater than that of a second polarizedchannel, the first polarized channel is before the second polarizedchannel in the first polarization weight vector.
 17. A non-transitorycomputer readable storage medium comprising instructions which, whenexecuted by a computer, cause the computer to perform operationsincluding: determining N to-be-encoded bits, wherein the N to-be-encodedbits comprise information bits and frozen bits, and N is a positiveinteger; obtaining a first polarization weight vector comprisingpolarization weights of N polarized channels, wherein the polarizationweights are a basis for information bit selection, and the Nto-be-encoded bits correspond to the N polarized channels; determiningpositions of the information bits based on the first polarization weightvector; and performing polar encoding on the N to-be-encoded bits toobtain polar-encoded bits.
 18. The computer storage medium according toclaim 17, wherein the obtaining a first polarization weight vectorcomprises: calculating the polarization weights W_(i) of the N polarizedchannels according to the following formula to obtain the firstpolarization weight vector:${W_{i} = {\sum\limits_{j = 0}^{n - 1}{B_{j}*( {2^{j} + \varphi} )^{\alpha}}}},$wherein i

B_(n−1)B_(n−2) . . . B₀, i is a channel index number, B_(n−1)B_(n−2) . .. B₀ is a binary representation of i, B_(n−1) is a most significant bit,B₀ is a least significant bit, B_(j)ϵ{0,1}, jϵ{0,1, . . . ,n−1}, iϵ{0,1,. . . ,N−1}, N=2^(n), ϕ and α are parameters that are preset based onthe target code length and a code rate of the polar code, and n is apositive integer.
 19. The computer storage medium according to claim 18,wherein ϕ=0, and α=¼.
 20. The computer storage medium according to claim17, wherein when the polarized channels in the first polarization weightvector are sorted in descending order of polarization weights, and aminimum sequence number value of the polarized channels is 1, if N=16,the first polarization weight vector is the following polarizationweight vector: 16,15,14,12,8,13,11,10,7,6,4,9,5,3,2,1; or if N=32, thefirst polarization weight vector is the following polarization weightvector:32,31,30,28,24,16,29,27,26,23,22,15,20,14,12,25,8,21,19,13,18,11,10,7,6,4,17,9,5,3,2,1:or if N=64, the first polarization weight vector is the followingpolarization weight vector:64,63,62,60,56,48,61,32,59,58,55,54,47,52,46,31,44,30,57,40,28,53,24,51,45,50,16,43,29,42,39,27,38,26,23,36,22,49,15,20,14,41,12,37,25,8,35,21,34,19,13,18,11,10,7,6,33,4,17,9,5,3,2,1; or if N=128, the first polarization weight vector is thefollowing polarization weight vector:128,127,126,124,120,112,125,96,123,122,119,64,118,111,116,110,95,108,94,121,104,92,63,117,62,88,115,109,60,114,80,107,56,93,106,103,91,48,102,90,61,87,100,32,59,86,113,79,58,84,55,78,105,54,76,47,101,52,89,46,72,99,31,44,85,98,30,57,83,40,77,28,82,53,75,24,51,74,45,71,50,16,43,70,97,29,42,68,39,27,81,38,26,23,36,73,22,49,15,69,20,14,41,67,12,66,37,25,8,35,21,34,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=256, thefirst polarization weight vector is the following polarization weightvector:256,255,254,252,248,240,253,224,251,250,247,192,246,239,244,238,128,223,236,222,249,232,220,191,245,190,216,243,237,188,242,127,208,235,184,126,221,234,231,124,219,176,230,218,189,120,215,228,160,187,214,241,112,207,186,212,183,125,206,233,182,96,123,204,175,229,180,122,217,174,119,200,227,64,159,118,172,213,226,111,158,185,116,211,168,110,205,156,210,181,95,108,203,152,179,94,121,202,173,104,199,178,92,63,144,117,171,198,225,62,157,88,170,115,196,167,109,60,155,114,209,166,80,107,154,56,151,164,93,106,201,103,150,177,91,48,143,102,197,148,90,61,142,87,169,100,195,32,59,86,140,113,194,165,79,58,153,84,55,136,163,78,105,54,149,162,76,47,101,52,147,89,46,141,72,99,146,31,44,85,139,98,193,30,57,138,83,40,135,77,28,82,53,134,161,75,24,51,132,74,45,71,50,145,16,43,70,97,29,42,137,68,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1; or if N=512, the firstpolarization weight vector is the following polarization weight vector:512,511,510,508,504,496,509,480,507,506,503,448,502,495,500,494,384,479,492,478,505,488,476,447,501,256,446,472,499,493,444,498,383,464,491,440,382,477,490,487,380,475,432,486,474,255,445,376,471,484,254,416,443,470,497,368,463,252,442,468,439,381,462,489,248,438,352,379,460,431,485,436,378,473,240,430,375,456,483,253,320,415,374,428,469,482,367,224,414,251,441,372,467,424,366,461,250,412,466,247,437,351,364,459,192,246,408,435,350,377,458,239,429,360,455,244,434,348,238,319,400,373,427,454,481,128,223,318,413,236,344,426,371,452,423,365,222,249,316,411,370,465,232,422,336,363,220,410,191,245,312,407,420,349,362,457,190,359,216,406,243,433,347,237,304,399,358,188,453,242,404,346,127,208,317,398,235,343,425,356,451,184,126,221,288,234,315,342,396,369,450,231,421,335,124,219,314,409,176,340,230,311,392,419,334,361,218,189,120,215,310,405,228,418,332,303,160,357,187,214,241,308,403,345,112,207,302,397,328,186,355,212,402,183,125,206,287,233,300,341,395,354,449,182,96,286,123,204,313,394,175,339,229,296,391,180,333,122,217,284,174,338,119,200,309,390,227,417,331,64,159,118,172,213,280,226,307,388,330,111,301,158,327,185,116,211,306,401,168,110,205,272,299,326,156,353,210,181,95,285,108,203,298,393,324,295,152,179,94,121,202,283,173,337,104,199,294,389,178,92,282,63,144,117,171,198,279,225,292,387,329,62,157,88,170,278,115,196,305,386,167,109,271,60,325,155,114,209,276,166,80,270,107,297,154,323,56,151,164,93,106,201,268,322,103,293,150,177,91,281,48,143,102,197,264,291,148,90,61,142,87,169,277100,195,290,385,32,59,86,140,113,194,275,165,79,269,58,153,84,274,55,136,163,78,105,267,321,54,149,162,76,266,47,101,263,52,147,89,46,141,72,262,99,289146,31,44,85,139,98,193,260,30,57,138,83,273,40,135,77,28,82,53,134,161,75,265,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,53,2,1; or if N=1024, the firstpolarization weight vector is the following polarization weight vector:1024,1023,1022,1020,1016,1008,1021,992,1019,1018,1015,960,1014,1007,1012,1006,896,991,1004,990,1017,1000,988,959,1013,768,958,984,1011,1005,956,1010,895,976,1003,952,894,989,1002,999,512,892,999,512,892987,944,998,986,767,957,888,983,996,766928,955,982,1009880,975,764,954,980,951,893,974,1001,760,950,864,511,891,972,943,997,948,510,890,985,752,942,887,968,995,508,765,832,927,886,940,981,994,879,736,926,763,953,504,884,979,936,878,973,762,924,978,759,949,863,496,876,971,704,758,920,947,862,509,889,970,751,941,872,967,756,946,480,860,507,750,831,912,885,939,966,993,506,640,735,830,925,748,856,938,503,883,964,935,877,734,448,761,828,923,502,882,977,744,934,848,495,875,732,922,703,500,757,824,919,932,861,494,874,969,702,871,728,918,755,945,384,479,859,492,749,816,911,870,700,965,754,916,478,858,505,639,720,829,910,747,855,937,488,868,963,696,476,638,733,800,447,746,827,854,908,501,881,962,743,933,847,256,446,636,731,826,921,472,688,852,499,742,823,904,931,846,493,873,730,444,701,498,632,727,822,917,740,930,383,464,844,491,815,672,869,699,726,440,753,820,915,382,477,857,490624,719,814,909,840,698,487,867,724,914,695,380,475,637,718,799,432,745,812,853,907,486,866,961,694,474,608,255,798,445,635,716,825,376,906,471,687,851,484,741,808,903,692,845,254,416,634,729,796,443,470,686,850,497,631,712,821,902,739,929,368,463,843,252,442,576,671,468,630,684,725,792,439,738,819,900,381,462,842,489,623,813,670,839,697,248,438,628,723,818,913,352,680,379,460,622,717,784,431,811,838,668,485,865,722,436,693,378,473,607,240,797,430,620,715,810,375,905,456,836,483,807,664,691,606,253,320,415,633,714,795,374,428,469,685,849,482,616,711,806,901,690,367,224,414,604,251,794,441,575,656,372,467,629,683,710,791,424,737,804,899,366,461,841,250,412,574,669,466,600,682,247,790,437,627,708,817,898,351,679,364,459,621,783,192,572,837,667,246,408,626,721,788,435,350,678,377,458,592,239,782,429,619,809,360,666,455,835,244,434,568,663,348,676,605,238,319,400,618,713,780,373,427,454,834,481,615,805,662,689,128,223,318,413,603,236,793,344,426,560,655,371,452,614,709,776,423,803,660,365,222,602,249,316,411,573,654,370,465,599,681,232,789,422,612,707,802,897,336,363,220,410,544,191,571,598,652,245,312,407,625,706,787,420,349,677,362,457,591,781,190,570,359,665,216,406,596,243,786,433,567,648,347,675,590,237,304,399,617,779,358,188,453,833,242,404,566,661,346,674,127,208,317,398,588,235,778,343,425,559,356,451,613,775,184,564,659,126,221,288,601,234,315,342,396,558,653,369,450,584,231,774,421,611,801,658,335,124,219,314,409,543,176,340,556,597,651,230,311,392,610,705,772,419,334,361,218,542,189,569,120,650,215,310,405,595,228,785,418,552,647,332,589,303,160,540,357,187,214,594,241,308,403,565,646,345,673,112,207,302,397,587,777,328,186,355,212,402,536,183,563,644,125,206,287,586,233,300,341,395,557,354,449,583,773,182,562,657,96,286,123,204,313,394,528,175,339,555,582,229,296,391,609,771,180,333,122,217,284,541,174,338,554,119,649,200,309,390,580,227,770,417,551,331,64,159,539,118,172,213,280,593,226,307,388,550,645,330,111,301,158,538,327,185,116,211,306,401,535,168,548,643,110,205,272,585,299,326,156,353,210,534,181,561,642,95,285,108,203,298,393,527,324,581,295,152,532,179,94,121,202,283,526,173,337,553,104,199,294,389,579,769,178,92,282,63,144,524,117,171,198,279,578,225,292,387,549,329,62,157,537,88,170,278,115,196,305,386,520,167,547,109,271,60,325,155,114,209,276,533,166,546,641,80,270,107,297,154,323,56,151,531,164,93,106,201,268,525,322,103,293,150,530,177,91,281,48,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,32,59,86,140,113,194,275,518,165,545,79,269,58,153,84,274,55,136,516,163,78,105,267,321,54,149,529,162,76,266,47,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,34,514,61,75,265,24,51,132,74,45,75,21,261,50,145,24,51,132,7445,71,26150,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2, 1; or if N=2048,the first polarization weight vector is the following polarizationweight vector:2048,2047,2046,2044,2040,2032,2045,2016,2043,2042,2039,1984,2038,2031,2036,2030,1920,2015,2028,2014,2041,2024,2012,1983,2037,1792,1982,2008,2035,2029,1980,2034,1919,2000,2027,1976,1918,2013,2026,2023,1536,1916,2011,1968,2022,2010,1791,1981,1912,2007,2020,1790,1952,1979,2006,2033,1904,1999,1788,1978,2004,1975,1917,1998,2025,1024,1784,1974,1888,1535,1915,1996,1967,2021,1972,1534,1914,2009,1776,1911,1992,2019,1532,1789,1856,1951,1910,1964,2005,2018,1903,1760,1950,1787,1977,1528,1908,2003,1960,1902,1997,1786,1948,1023,2002,1783,1973,1887,1520,1900,1995,1728,1022,1782,1944,1971,886,1533,1913,1994,1775,1965,1896,199020,1780,1504,188,4531,17,7774,855,1936,1909,1963,1990,2017,1016,1530,1664,1759,1854,1949,1772,1880,1962,1527,1907,1988,1959, 1901,1758,1472,1785, 1852,1947,1526, 1906,1008,2001,1768,1958,1872, 1519, 1899,1756,1946,1727,1021,1524,1781,1848,1943,1956,1885,1518,1898,1993,1726,99292,1895,1752,1019,1942,1779,1969,1408,1503,1883,1516,1773,1840,1935,1894,1724,1989,1018,1778,1940,1502,1015,1882,1529,1663,1744,1853,1934,1771,1879,1961,1512,1892,1987,960,1720,1014,1500,1662,1757,1824,1471,1770,1851,1878,1932,1525,1905,1007,1986,1767,1957,1871,1012,1280,1470,1660,1755,1850,1945,1496,1712,1876,1006,1523,1766,1847,1928,1955,1870,1517,1897,1754,1468,1725,896,991,1522,1656,1751,1846,1004,1941,1764,1954,1407,1488,1868,1515,1839,1696,990,1893,1723,1750,1017,1464,1777,1844,1939,1406,1501,1000,1881,1514,1648,1743,1838,1933,1864,1722,1511,988,1891,1748,959,1938,1719,1013,1404,1499,1661,1742,1823,1456,1769,1836,1877,193,1510,1890768,1985,9581718984,1498,1632,1011,1279,1822,1469,1659,1740,1849,1400,1930,1495,1711,1875,1005,1508,1765,1832,1927,956,1716,1869,2781,440,1658,1753,1820,1467,1494,1710,895,1874,97689,587,497,1521,1655,1736,1845,1003,1926,1763,1953,1392,1487,1867,952,1276,1466,1600,1695,894,989,1492,1654,1708,1749,1816,1002, 1463,1762, 1843,1924,1405,1486,999,1866, 1513,1647,1837,1694,512, 1863, 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orif N=4096, the first polarization weight vector is the followingpolarization weight 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2,1226,1307,2341,739,1550,2198,929,1197,368,1361,2578,463,1577,2225,1128,843,1223,1318,252,1413,1603,2139,442,576,1793,2329,671,1202,2096,3089,468,630,2191,684,725,792,1116,2571,2150,439,1306,2245,2312,738,819,1087,2625,900,1168,381,2339,462,1548,2196,1141,1195,842,1222,1303,489,623,1602,2138,813,1249,1316,670,1411,1573,2109,839,2190,697,248,1353,2570,2135,438,628,2217,2325,723,818,1086,2148,913,1181,352,2243,2338,680,1561,2433,1112,2567,1194,379,1302,3081,460,622,1139,717,784,1220,1329,431,1410,1544,2080,811,1191,838,668,485,1571,2107,2134,865,2188,1133,722,1295,2161,436,2242,2323,693,1084,1349,2566,1179,378,2213,473,607,1138,2593,240,797,1233,1300,2127,430,620,1557,2317,715,810,1190,375,905,3077,456,1570,2106,1665,8362201,1104,483,1294,2132,807,1131,2322,664,1321,2103,1178,2184,691,606,1347,2564,253,320,2211,2126,415,633,1080,714,795,1175,374,2153,428,469,1555,2315,685,1188,849,1117,3075,482,616,2369,711,806,1130,901,1,1225,1292,2102,1549,2197,690,367,1346,2577,224,2210,1127,414,604,1317,251,794,1174,2124,441,575,1554,2314,656,1201,372,2095,3074,467,629,683,710,791,1115,2149,424,1305,2311,737,804,1072,899,1167,366,2100,461,1547,2195,1126,841,1221,1288,250,1601,2137,412,574,1315,669,172,2094,466,600,2189,682,247,790,1114,2569,2120,437,627,2310,708,817,1085,2147,898,1166,351,2337,679,1546,2194,1111,1193,364,1301,459,621,1124,783,1219,1314,192,1409,1543,2079,572,837,667,246,2092,2133,408,626,2187,721,788,1056,2146,435,350,2241,2308,678,1083,1110,2565,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,2078,809,1189,360,666,455,1569,2105,835,2186,1103,244,1293,2131,434,568,2321,663,1082,2088,1177,348,2183,676,605,1108,2563,238,319,1298,2125,400,618,1079,713,780,1160,373,427,454,1540,2076,1187,834,1102,481,615,2130,805,1129,662,1291,2101,2182,689,128,1345,2562,223,318,2209,413,603,1078,236,793,1173,344,2123,426,560,1553,2313,655,1186,371,3073,452,614,709,776,1100,423,1290,2072,803,1071,660,365,2099,222,2180,1125,602,1287,249,316,2122,411,573,1076,654,1171,370,2093,465,599,681,232,789,1113,2119,422,612,2309,707,802,1070,897,1165,336,2098,1545,2193,1096,363,1286,220,1123,410,544,1313,191,1170,2064,571,598,652,245,312,2091,2118,407,625,706,787,1055,2145,420,349,2307,677,1068,1109,1163,362,457,591,1122,781,1217,1284,190,154412077,570,359,665,216,2090,406,596,2185,243,786,1054,2116,433,567,2306,648,1081,2087,1162,347,675,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,2075,833,1101,242,2129,404,566,661,1052,2086,346,2181,674,127,1106,2561,208,317,398,588,1077,235,778,1158,343,425,559,1538,2074,1185,356,451,613,775,1099,184,1289,2071,564,659,126,2084,221,288,2179,601,1048,234,315,342,2121,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,2070,801,1069,658,335,2097,2178,1095,124,1285,219,314,409,543,1074,176,1169,340,2063,556,597,651,230,311,2117,392,610,705,772,1040,419,334,2068,1067,1094,361,218,1121,542,1283,189,2062,569,120,650,215,310,2089,405,595,228,785,1053,2115,418,552,2305,647,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,2060,594,241,308,2114,403,565,646,1051,2085,345,673,112,1105,207,302,397,587,1062,7777,1157,328,186,1537,2073,355,212,402,536,183,1050,2056,563,644,125,2083,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,2069,562,657,96,2082,286,2177,123,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,180,333,2067,1093,122,217,284,541,1044,174,338,2061,554,119,649,200,309,390,580,227,770,1038,417,551,2066,1065,331,1091,64,1281,159,539,118,172,213,280,2059,593,226,307,2113,388,550,645,1036,330,111,1090,301,158,1061,538,327,185,2058,116,211,306,401,535,168,1049,2055,548,643,110,205,272,585,1032,299,326,156,1059,353,210,534,181,2054,561,642,95,2081,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179,94,2052,121,202,283,526,1043,173,337,553,104,199,294,389,579,769,1037,178,2065,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,549,1035,329,1089,62,157,537,88,170,278,2057,115,196,305,386,520,167,1034,547,109,271,1031,60,325,155,114,209,276,533,166,2053,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,2051,106,2001,268,525,1028,322,103,293,150,530,177,2050,91,281,48,1041,143,523,102,197,264,577,291,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,545,79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,2049,76,266,47,1026,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,95,3,2,1; or when a minimum sequence number in the first polarizationweight vector is 0, 1 is subtracted from sequence numbers in the firstpolarization weight vector successively.